Valley Functions Overview

• Valley functions are a family of technically distinct constructs that isolate fine structures hidden by coarser descriptions across domains.
• They are applied in valleytronics, metamaterials, semiconvex analysis, combinatorial generating functions, and Morse theory, each with domain-specific definitions.
• Examples include valley-resolved currents, nonlinear response functions, valley transforms for singularity detection, and local geometric models in Morse theory.

“Valley functions” does not denote a single standardized object. In the arXiv literature surveyed here, it names several distinct constructions: valley-resolved observables attached to inequivalent $K/K'$ sectors in condensed-matter and metamaterial systems; nonlinear and stochastic response functions for valley polarization dynamics; compensated-convex valley transforms for semiconvex analysis; valley generating functions in algebraic combinatorics; and local “val” structures in Morse theory (Das et al., 2023, Zhang et al., 2016, Shin, 12 Jun 2026, Marin, 11 Sep 2025). A plausible common motif is that each usage isolates structure hidden by a coarser description, but the underlying objects, equations, and applications are domain-specific.

1. Terminological domains

Across the cited literature, the term is used in several non-equivalent ways. This suggests that “valley functions” is best treated as a family of technical meanings rather than a single concept (Das et al., 2023, Zhang et al., 2016, Semina et al., 2022, Shin, 12 Jun 2026, Marin, 11 Sep 2025).

Domain	Valley function	Representative role
Valleytronics and nonlinear transport	$j_a^{\rm V}$, $\sigma_{ab}^{\rm V}$, $\chi_{a;bc}^{\rm NLV}$	Valley-resolved current and conductivity
Excitonic and trionic dynamics	$P(t)$, $P(E)$, $S_z(t)$, autocorrelation and switching functions	Valley polarization, relaxation, bistability
Semiconvex analysis	$V_\lambda(f)=C_\lambda^u(f)-f$	Singular-set extraction
Algebraic combinatorics	$\mathrm{Val}_{n,k}(X;q,t)$	Labelled Dyck-path generating function
Morse theory	“fonction val” on a tubular neighborhood	Canonical local model and controlled deformations

Only the condensed-matter, photonic, phononic, and excitonic usages refer to Brillouin-zone valleys $K$ and $j_a^{\rm V}$0. The analytical, combinatorial, and Morse-theoretic usages are independent of momentum-space valley physics and instead use “valley” in a geometric or combinatorial sense.

2. Polarization, relaxation, and valley splitting

In monolayer $j_a^{\rm V}$1, valley functions were identified with the time- and energy-dependent polarization of trion branches, $j_a^{\rm V}$2 and $j_a^{\rm V}$3. The valley polarization is extracted from co- and cross-circular differential transmission as

$j_a^{\rm V}$4

Under resonant or quasi-resonant trion excitation, the inter-valley branch obeys

$j_a^{\rm V}$5

whereas the intra-valley branch satisfies

$j_a^{\rm V}$6

with no observable decay over $j_a^{\rm V}$7. The long-lived response is specific to resonantly created intra-valley trions; off-resonant excitation at the exciton resonance instead inherits the exciton’s $j_a^{\rm V}$8 depolarization and the 1–2 ps trion-formation timescale, thereby masking the intrinsic trion valley behavior (Singh et al., 2016).

A separate dynamical use appears in the theory of excitons in atom-thin semiconductors, where the primary valley order parameter is

$j_a^{\rm V}$9

The polarization-dependent relaxation law is

$\sigma_{ab}^{\rm V}$0

so the effective magnetic field generated by exciton–exciton interactions slows valley relaxation as $\sigma_{ab}^{\rm V}$1 grows. Under cw pumping, the steady state satisfies

$\sigma_{ab}^{\rm V}$2

leading to a cubic equation and bistability when $\sigma_{ab}^{\rm V}$3. Near a stable branch, the valley-noise autocorrelation function is

$\sigma_{ab}^{\rm V}$4

and noise-induced switching follows a Kramers-like law

$\sigma_{ab}^{\rm V}$5

In that framework, “valley functions” includes the deterministic drift, the steady-state response, the autocorrelation function, the quasi-potential, and the switching rate (Semina et al., 2022).

In centrosymmetric monolayer $\sigma_{ab}^{\rm V}$6, valley functions are controlled by weak valley-layer coupling. The valence-band valley splitting obeys

$\sigma_{ab}^{\rm V}$7

with $\sigma_{ab}^{\rm V}$8 and $\sigma_{ab}^{\rm V}$9. For out-of-plane magnetization, the response is linear in $\chi_{a;bc}^{\rm NLV}$0, with $\chi_{a;bc}^{\rm NLV}$1 at $\chi_{a;bc}^{\rm NLV}$2; for in-plane magnetization, the coupling vanishes and $\chi_{a;bc}^{\rm NLV}$3. Reversing only $\chi_{a;bc}^{\rm NLV}$4 or only $\chi_{a;bc}^{\rm NLV}$5 switches the valley polarization, whereas reversing both simultaneously leaves it unchanged. In the simply stacked bilayer, the $\chi_{a;bc}^{\rm NLV}$6-antiferromagnetic state yields spontaneous valley polarization without external electric field, with $\chi_{a;bc}^{\rm NLV}$7 for AA stacking and $\chi_{a;bc}^{\rm NLV}$8 for AB stacking (Guo et al., 28 May 2025).

3. Transport coefficients, filtering, and switching in electronic systems

A formal transport-theoretic definition is given by valley-resolved response coefficients. The linear valley Hall current is

$\chi_{a;bc}^{\rm NLV}$9

and the nonlinear counterpart is

$P(t)$0

For systems with both inversion and time-reversal symmetry, the ordinary Berry curvature vanishes pointwise, so the linear valley Hall current and the Berry-curvature-dipole contribution vanish. The surviving second-order response is instead generated by the electric-field-modified Berry curvature through the Berry connection polarizability tensor. In tilted massless Dirac fermions,

$P(t)$1

so $P(t)$2. For $P(t)$3, $P(t)$4, $P(t)$5, $P(t)$6, and $P(t)$7, the nonlinear valley Hall angle is estimated as $P(t)$8. The predicted nonlocal signal scales as

$P(t)$9

which differs from both no-valley-Hall and linear-valley-Hall scaling laws (Das et al., 2023).

Several device papers use “valley functions” operationally, as filtering, splitting, or source behavior. In twisted bilayer graphene, a smooth electrostatic barrier that drives a Lifshitz transition across the junction acts as a valley splitter: one minivalley is transmitted and the other reflected, with $P(E)$0 and $P(E)$1 in the filter regime. In a six-terminal geometry, $P(E)$2 and $P(E)$3 is negligible, demonstrating zero-field transverse valley focusing (Beule et al., 2019). In a gate-defined $P(E)$4 junction, small band offset yields a valley filter dominated by intravalley scattering, while large band offset yields a valley source dominated by intervalley scattering; the filter polarity is antisymmetric under $P(E)$5 and vanishes after angular integration by time-reversal symmetry, whereas the source generates nonzero angle-integrated outward valley flux. In TMDC cross-bar collection, the reported collected valley polarization exceeds $P(E)$6 for $P(E)$7 and $P(E)$8 (Tu et al., 2017). In gapped bilayer graphene, a gate-defined splitter measures

$P(E)$9

which equals the injected valley polarization for a perfect splitter with unsaturated collector arms, and the design remains tolerant to gate misalignment $S_z(t)$0–$S_z(t)$1 for typical $S_z(t)$2 and $S_z(t)$3 (Luna et al., 2024).

All-optical and synchronized-gate control realize a related set of transport functions. In pristine graphene irradiated by a bicircular counter-rotating drive, the relative phase $S_z(t)$4 controls valley-selective quasi-energy gaps and Floquet occupations. The valley-filter polarization

$S_z(t)$5

reaches $S_z(t)$6 near $S_z(t)$7 and $S_z(t)$8, while two driven segments in series form a valley valve whose on/off state is controlled by $S_z(t)$9 (Kumari et al., 2024). In graphene/hBN, synchronized square-wave tip and bias voltages of the same polarity generate periodically modulated pure valley currents with $V_\lambda(f)=C_\lambda^u(f)-f$0 during the ON half-cycles. The reported peak currents are $V_\lambda(f)=C_\lambda^u(f)-f$1 for $V_\lambda(f)=C_\lambda^u(f)-f$2 and $V_\lambda(f)=C_\lambda^u(f)-f$3 for $V_\lambda(f)=C_\lambda^u(f)-f$4; reversing the synchronized gate polarity swaps the ON and OFF valley channels (Belayadi et al., 27 May 2025).

4. Non-Hermitian and chiral edge implementations

In lossy phononic metamaterials, valley functions are realized through non-Hermitian bandstructure and boundary localization. The Bloch Hamiltonian is

$V_\lambda(f)=C_\lambda^u(f)-f$5

with purely lossy intracell coupling. Near a valley,

$V_\lambda(f)=C_\lambda^u(f)-f$6

so the decay rate is valley- and direction-dependent. This generates three distinct functions: a valley filter through valley-resolved nonreciprocity, a valley-dependent skin effect through opposite spectral windings at $V_\lambda(f)=C_\lambda^u(f)-f$7 and $V_\lambda(f)=C_\lambda^u(f)-f$8, and valley-projected edge states with boundary-dependent lifetimes. Experimentally, a phononic ABA sandwich shows hybrid $V_\lambda(f)=C_\lambda^u(f)-f$9 input and nearly pure $\mathrm{Val}_{n,k}(X;q,t)$0 output at $\mathrm{Val}_{n,k}(X;q,t)$1; the same platform exhibits boundary-resolved skin accumulation and anomalous beam splitting with transmission only to the $\mathrm{Val}_{n,k}(X;q,t)$2 port (Yin et al., 24 Oct 2025).

A topological photonic counterpart is furnished by chiral valley edge states. The low-energy description is

$\mathrm{Val}_{n,k}(X;q,t)$3

When the interface masses satisfy $\mathrm{Val}_{n,k}(X;q,t)$4 for only one valley, the resulting edge mode is both chiral and valley-confined. In hybrid Chern-photonic-crystal and valley-photonic-crystal interfaces, this enables valley multiplexing, demultiplexing, and valley-locked waveguide crossings. The demonstrated frequency window is approximately $\mathrm{Val}_{n,k}(X;q,t)$5–$\mathrm{Val}_{n,k}(X;q,t)$6, and the crossing device exhibits simulated crosstalk below $\mathrm{Val}_{n,k}(X;q,t)$7 for port-1 excitation and below $\mathrm{Val}_{n,k}(X;q,t)$8 for port-3 excitation; the measured values are approximately $\mathrm{Val}_{n,k}(X;q,t)$9 and $K$0, respectively (Liu et al., 20 May 2025).

5. Valley transforms in semiconvex analysis

In analysis, the valley function is the valley transform associated with compensated convex transforms (Zhang et al., 2016). For a function $K$1 of linear growth,

$K$2

$K$3

and the ridge, valley, and edge transforms are

$K$4

$K$5

The paper emphasizes that this valley transform is always non-negative and differs by sign from an earlier convention.

The central theorem identifies singular points of a locally semiconvex function with scale $K$6-valley points. If $K$7 is singular, then

$K$8

where $K$9 is the radius of the minimal bounding sphere of the Fréchet subdifferential $j_a^{\rm V}$00. At differentiable points,

$j_a^{\rm V}$01

The limit

$j_a^{\rm V}$02

is therefore a scale $j_a^{\rm V}$03-valley landscape function of the singular set. Equivalently, the upper transform has the asymptotic expansion

$j_a^{\rm V}$04

at singular points.

For locally semiconvex functions with linear modulus, the theory also yields a first-order geometric descriptor: $j_a^{\rm V}$05 where $j_a^{\rm V}$06 is the center of the minimal bounding sphere of $j_a^{\rm V}$07. For DC-functions $j_a^{\rm V}$08, the scale $j_a^{\rm V}$09-edge transform satisfies

$j_a^{\rm V}$10

Worked examples include $j_a^{\rm V}$11, for which $j_a^{\rm V}$12, as well as Euclidean distance and squared-distance functions, where the ridge or valley limits recover medial-axis geometry. The locality property

$j_a^{\rm V}$13

shows that the convex envelopes needed for $j_a^{\rm V}$14 and $j_a^{\rm V}$15 can be computed in a shrinking neighborhood of $j_a^{\rm V}$16, which is important for numerical implementations.

6. Valley generating functions and Morse-theoretic valleys

In algebraic combinatorics, the valley function is a labelled-Dyck-path generating series. For a $j_a^{\rm V}$17-decorated labelled path $j_a^{\rm V}$18, with $j_a^{\rm V}$19 defined from primary and secondary attacks and $j_a^{\rm V}$20 a set of decorated contractible valleys, the valley generating function is

$j_a^{\rm V}$21

with fixed-diagonal-multiset slices

$j_a^{\rm V}$22

The cited result proves that the slices $j_a^{\rm V}$23 and, for $j_a^{\rm V}$24, $j_a^{\rm V}$25 are symmetric; equivalently, the coefficients of $j_a^{\rm V}$26 and $j_a^{\rm V}$27 in $j_a^{\rm V}$28 are symmetric functions. The proof is based on an adjacent exchange identity for scaffold classes and an operator calculus built from two commuting half-twists, with

$j_a^{\rm V}$29

and the companion symmetric identity

$j_a^{\rm V}$30

for the area-$j_a^{\rm V}$31 cases (Shin, 12 Jun 2026).

A different geometric meaning appears in Morse theory, where a “fonction val” is a tubular neighborhood of a submanifold $j_a^{\rm V}$32 on which the function is fiberwise a translated metric. If $j_a^{\rm V}$33 is the tube projection and $j_a^{\rm V}$34 the fiber metric, then

$j_a^{\rm V}$35

and on the tube boundary $j_a^{\rm V}$36 factorizes through the crest $j_a^{\rm V}$37. This structure supports an “ascenseur de val” that changes $j_a^{\rm V}$38 while keeping the crest fixed, a canonical Morse normal form

$j_a^{\rm V}$39

and a cancellation theorem that reduces elimination of a pair of critical points to a one-dimensional cubic model

$j_a^{\rm V}$40

The paper’s stated purpose is precisely to obtain canonical form, moving critical values, and elimination of pairs of critical points “without gluing technicities,” by working with valley functions on smooth manifolds and using Moser’s path method (Marin, 11 Sep 2025).

Taken together, these usages show that “valley functions” names a technical pattern rather than a universal object. In valleytronics and metamaterials, the term centers on valley-resolved currents, conductivities, polarizations, and edge-mode transfer characteristics; in analysis it denotes a multiscale singularity detector derived from compensated convex transforms; in algebraic combinatorics it is a decorated-path generating function; and in Morse theory it is a local geometric model for controlled deformation of smooth functions.