Frequency-Momentum Winding Numbers

• Frequency-momentum winding numbers are topological invariants defined as mapping degrees of det F and its ω-derivatives that classify nonlinear exceptional points in m-band systems.
• They refine conventional linear theories by revealing an integer topology in PT-symmetric EP2’s and extending classifications under PT, CP, and chiral symmetries.
• Their construction implies a doubling theorem where exceptional points occur in oppositely charged pairs, ensuring the global topological charge cancels across the Brillouin zone.

to=arxiv_search.search 天天中彩票彩金asydict 微信上的天天中彩票 code: {"query":"(Yoshida, 1 Apr 2026) frequency-momentum winding numbers exceptional points", "max_results": 5} to=arxiv_search.search 彩彩票娱乐 code: {"query":"frequency-momentum winding numbers non-Hermitian arXiv", "max_results": 10} Frequency-momentum winding numbers are topological invariants for exceptional points in nonlinear eigenvalue problems, introduced to characterize nonlinear EP$n$s in $m$-band systems throughout the Brillouin zone for arbitrary $n$ and $m$ with $m\geq n$. In the formulation developed in "Nonlinear Frequency-Momentum Topology and Doubling of Multifold Exceptional Points" (Yoshida, 1 Apr 2026), they are integer-valued mapping degrees built from $\det F(\omega,\mathbf{k})$ and its $\omega$-derivatives, and they provide a unified proof of the doubling theorem for multifold exceptional points in the absence of symmetry and under several symmetry constraints, including $PT$ and charge-conjugation-parity symmetries. The same construction also refines the linear theory: even in the linear limit, it indicates a $\mathbb{Z}$ topology for $PT$-symmetric EP$m$0s beyond the previously reported $m$1 topology, and it extends to a class of coupled resonators in which nonlinearity enters via the eigenvectors while the spectrum is determined by a nonlinear scalar equation for the frequency (Yoshida, 1 Apr 2026).

1. Nonlinear spectral setting and local exceptional-point structure

The construction begins from a nonlinear eigenvalue problem

$m$2

with characteristic polynomial $m$3. A regularity condition is assumed at large frequency: $m$4 becomes independent of $m$5 for large $m$6, as exemplified by $m$7 for $m$8 with integer $m$9 (Yoshida, 1 Apr 2026).

An EP$n$0 is a point in complex frequency-momentum space at which $n$1 eigenvalues, and generically the corresponding eigenvectors, coalesce. Locally, the dispersion exhibits the fractional-power branching characteristic of non-Hermitian degeneracies: $n$2 so encircling the singularity cycles the spectrum through $n$3 Riemann sheets. The paper states that an EP$n$4 emerges at $n$5 if and only if

$n$6

with $n$7 (Yoshida, 1 Apr 2026).

This formulation is explicitly insensitive to eigenvalue labeling. That feature is central: the invariant is built directly from $n$8 and its derivatives, rather than from a chosen band decomposition. A common misconception is to identify these invariants with ordinary point-gap windings of individual bands. The 2026 construction is different in scope and target, because it is designed to treat nonlinear eigenvalue problems and higher-order EP$n$9s uniformly (Yoshida, 1 Apr 2026).

2. Mathematical construction of the winding numbers

Without symmetry, the EP$m$0 conditions are rewritten as the vanishing of a real vector $m$1 built from $m$2 and its $m$3-derivatives up to order $m$4: $m$5 The invariant is then the mapping degree of the normalized vector $m$6 from a small sphere $m$7 enclosing the EP$m$8 in $m$9 space to the target sphere $m\geq n$0 (Yoshida, 1 Apr 2026).

The paper defines the frequency-momentum winding number as

$m\geq n$1

with

$m\geq n$2

Its sign follows the standard mapping-degree convention: changing the orientation of the enclosing sphere flips the sign of $m\geq n$3 (Yoshida, 1 Apr 2026).

The construction applies to arbitrary $m\geq n$4 and to $m\geq n$5-band systems with $m\geq n$6 because it uses only $m\geq n$7 and its $m\geq n$8-derivatives. Different EP multiplicities are distinguished by the length of the $m\geq n$9-vector and the codimension of the singularity, which together determine the dimension $\det F(\omega,\mathbf{k})$0 of the enclosing sphere and the corresponding integer $\det F(\omega,\mathbf{k})$1 (Yoshida, 1 Apr 2026).

In this framework, the topological content of a nonlinear EP$\det F(\omega,\mathbf{k})$2 is encoded as a local degree of the map $\det F(\omega,\mathbf{k})$3. This suggests a direct generalization of familiar winding-number logic from one-dimensional point-gap topology to higher-codimension singularities in frequency-momentum space, but with the determinant-based construction replacing eigenvalue-tracking.

3. Symmetry-resolved formulations and classification

The paper gives symmetry-adapted $\det F(\omega,\mathbf{k})$4-vectors and domains of integration for several non-Hermitian symmetry classes. For $\det F(\omega,\mathbf{k})$5 symmetry, closed on real $\det F(\omega,\mathbf{k})$6, the constraints become $\det F(\omega,\mathbf{k})$7 real equations with

$\det F(\omega,\mathbf{k})$8

and the integration sphere lies in the real-frequency subspace. For $\det F(\omega,\mathbf{k})$9 symmetry, closed at $\omega$0, derivative-parity constraints determine which $\omega$1-derivatives enter the $\omega$2-vector. For combined $\omega$3, also closed at $\omega$4 with $\omega$5 real, only even or odd derivatives appear depending on parity. Chiral symmetry is rewritten as a $\omega$6-like condition on imaginary frequency by defining $\omega$7 with $\omega$8 (Yoshida, 1 Apr 2026).

The codimensions and classification groups stated in the paper are as follows.

Symmetry class	Codimension and domain	Classification
No symmetry	$\omega$9, $PT$0	$PT$1
$PT$2	$PT$3, $PT$4	$PT$5
$PT$6	$PT$7 for even $PT$8, $PT$9 for odd $\mathbb{Z}$0, $\mathbb{Z}$1	$\mathbb{Z}$2
$\mathbb{Z}$3	$\mathbb{Z}$4 for even $\mathbb{Z}$5, $\mathbb{Z}$6 for odd $\mathbb{Z}$7, $\mathbb{Z}$8, $\mathbb{Z}$9 real	$PT$0 for $PT$1; $PT$2 for $PT$3

For the low-order $PT$4 cases, the paper uses a $PT$5 invariant

$PT$6

with $PT$7 for $PT$8 and $PT$9 for $m$00 (Yoshida, 1 Apr 2026).

A central classification result is that even in the linear limit, $m$01-symmetric EP$m$02s carry a $m$03 topology under the frequency-momentum winding number. The paper explicitly contrasts this with the previously reported $m$04 topology, and its Supplemental numerics show pairs of $m$05-symmetric EP$m$06s with identical $m$07 that do not annihilate, which indicates a finer stability structure than parity-only classification (Yoshida, 1 Apr 2026).

4. Doubling theorem for multifold exceptional points

The principal global application of the invariant is the doubling theorem for EP$m$08s. In the absence of symmetry, EP$m$09s occur in $m$10-dimensional $m$11-$m$12 space, with codimension $m$13 and $m$14. Because the Brillouin zone is periodic and the only boundary contribution comes from $m$15, where $m$16 is $m$17-independent, the total winding vanishes: $m$18 The same boundary argument applies under $m$19, $m$20, and $m$21 symmetry. Consequently, any EP$m$22 with $m$23 must be accompanied by another EP$m$24 with $m$25 (Yoshida, 1 Apr 2026).

The topological mechanism is the additivity and homotopy invariance of the mapping degree. Since $m$26 is integer-valued and locally stable, the sum over all enclosed EP$m$27s can be converted to a boundary integral. The condition that $m$28 becomes $m$29-independent at large $m$30 kills all momentum-derivative contributions there, so the total charge over the Brillouin zone must vanish (Yoshida, 1 Apr 2026).

This result generalizes the familiar doubling theorem beyond two-fold exceptional points. Before this formulation, the topology of multifold exceptional points across the Brillouin zone lacked a general characterization, and the doubling theorem was essentially limited to two-fold exceptional points. The frequency-momentum winding numbers provide the missing invariant structure for arbitrary $m$31 and $m$32 (Yoshida, 1 Apr 2026).

The robustness statement is likewise explicit: small perturbations that do not close $m$33 and preserve the relevant symmetry do not change $m$34. Exceptional points may move, but their charges are unchanged unless oppositely charged singularities collide and annihilate. This suggests that the doubling theorem is stable under disorder and smooth deformations so long as the large-$m$35 asymptotic independence from $m$36 is preserved (Yoshida, 1 Apr 2026).

5. Linear limit, computation, and representative models

In the linear limit, the frequency-momentum invariant reduces to known discriminant-based indices for EP$m$37s. The Supplemental Material shows, for a $m$38 toy model, that the FM winding number $m$39 coincides with the discriminant winding, and states that this coincidence holds for an arbitrary model. At the same time, the $m$40-symmetric EP$m$41 case is enriched rather than merely reproduced, because the FM construction yields an integer $m$42 instead of only $m$43 parity (Yoshida, 1 Apr 2026).

The paper gives a practical computational procedure. One first locates EP$m$44 points by solving $m$45 with $m$46. One then constructs the symmetry-appropriate $m$47-vector, chooses a small sphere $m$48 around the singularity, evaluates $m$49 on a discretized mesh, and computes the mapping degree integral. The paper also gives a Hermitian mapping trick: define a Hermitian Hamiltonian $m$50 whose Chern or winding number equals the FM invariant, and then compute that topological index by standard lattice methods such as Fukui–Hatsugai–Suzuki (Yoshida, 1 Apr 2026).

Several representative models are discussed. One is a $m$51-symmetric EP$m$52 model in two dimensions with

$m$53

for which the figure shows EP$m$54 points carrying $m$55 and $m$56. Supplemental toy models of $m$57-symmetric EP$m$58s show two distinct behaviors: in a $m$59 model, two EP$m$60s each with $m$61 approach but do not annihilate as $m$62 is lowered, whereas in a $m$63 model EP$m$64 pairs do annihilate when the sum of their charges is zero (Yoshida, 1 Apr 2026).

The construction also extends to nonlinear systems in which the nonlinearity enters via eigenvectors rather than eigenvalues, provided the spectrum is determined by a scalar nonlinear equation. In the coupled-resonator example, the stable-state eigenvalues satisfy $m$65, and one simply sets $m$66. Near an EP$m$67, the paper obtains $m$68 from the corresponding $m$69-vector expansion (Yoshida, 1 Apr 2026).

A further practical advantage is that the method avoids branch-cut and Riemann-sheet bookkeeping. Because it depends on $m$70 and its derivatives, not on explicit eigenvalue labeling, it is gauge-invariant under basis changes and numerically stable when the integration sphere is varied without crossing other zeros of $m$71 (Yoshida, 1 Apr 2026).

6. Related meanings in adjacent literatures

The expression "frequency-momentum winding" also appears in several related but distinct settings. In one-dimensional non-Hermitian band theory, the standard point-gap invariant is the spectral winding of complex energy around a reference energy,

$m$72

or, in multiband form,

$m$73

For one-band finite-range one-dimensional models, nonzero spectral winding is equivalent to the non-Hermitian skin effect under open boundary conditions (Zhang et al., 2019). The 2026 FM invariant differs from this point-gap construction because it is formulated directly at the level of $m$74 and its $m$75-derivatives, and it does not require an isolated point gap or eigenvalue labeling (Yoshida, 1 Apr 2026).

In interacting topological insulators, a different frequency-space invariant arises from the Green's function $m$76. Under the local self-energy approximation, the interacting topological index factorizes into a momentum-space invariant and a frequency-domain winding number determined by the contour of $m$77; this frequency winding can destroy or multiply the momentum-space topological class without long-range order (Wang et al., 2011). In Weyl semimetals, a Green's-function 3-form on a closed $m$78 in $m$79 space gives a frequency-momentum winding equal to the monopole charge of the Weyl node and to the Berry-curvature Chern number up to the projection convention (Elbistan, 2016).

In one-dimensional chiral non-Hermitian systems, the winding number can be half-integer and admits a geometric decomposition into two integer windings $m$80 and $m$81 associated with two exceptional points, with the energy vorticity given by their difference. That construction explains why encircling only one exceptional point yields half-integer winding and why left and right zero-mode counts are separately controlled by $m$82 and $m$83 (Yin et al., 2018). A different experimental direction uses the momentum-space single-particle spectral function $m$84: in the models treated, topology is encoded in $m$85 crossings or $m$86 contours of the spectral weight, enabling extraction of winding or Chern numbers from ARPES or STS data (Estake et al., 2024).

A further source of potential confusion is the use of "momentum and winding" in T-duality. There the relevant exchange concerns fiberwise Fourier labels and tensor-power labels of line bundles on circle bundles with $m$87-flux, implemented by an exotic Hori transform; it is a geometric and cohomological exchange of momentum and winding numbers, not a non-Hermitian frequency-momentum topology of exceptional points (Han et al., 2017).

Taken together, these works show that "frequency-momentum winding" is not a single invariant with a universal definition. In the specific sense established in (Yoshida, 1 Apr 2026), however, frequency-momentum winding numbers are determinant-and-derivative mapping degrees that classify nonlinear exceptional points, control their global doubling across the Brillouin zone, and interpolate between nonlinear spectral topology and several earlier winding constructions in the linear and Green's-function-based literatures.