NM-Boost: Non-Hermitian Deformation & Dark Matter

• NM-Boost is a label that describes an imaginary boost deformation in integrable many-body systems, complexifying spectral parameters and inducing energy-twisted boundary conditions.
• It is derived by modifying the Hermitian boost flow—removing the factor of i—to uncover non-Hermitian effects such as point-gap topology and the skin effect in both free and interacting models.
• Separately, NM-Boost also denotes boosted dark matter in non-minimal sectors, where annihilation and semi-annihilation mechanisms produce relativistic dark particles detectable via terrestrial scattering.

NM-Boost is a label used in the supplied literature for more than one research object. In its most formal usage, it denotes the “imaginary boost deformation” introduced for integrable many-body systems: the familiar Hermitian boost flow generated by $B[H(\kappa)] = \sum_x x\,h_x$ and $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$ is made non-Hermitian by removing the factor of $i$, so that $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$. This deformation complexifies spectral parameters, is equivalent to energy-twisted boundary conditions, and produces complex spectral winding under periodic boundary conditions and the non-Hermitian skin effect under open boundary conditions (Guo et al., 2023). In separate dark-sector literature, the same label is also used for boosted dark matter phenomena in non-minimal or multi-component sectors, including annihilation and semi-annihilation mechanisms that generate relativistic dark particles detectable through scattering in terrestrial experiments (Agashe et al., 2014, Alhazmi et al., 2016, Fujiwara et al., 1 Jun 2026).

1. Terminological scope

In the supplied literature, “NM” appears with distinct meanings across subfields. The resulting ambiguity is substantive, because the same surface label is attached to unrelated mathematical constructions, dark-sector phenomenology, and heterostructure notation.

Usage	Meaning in the supplied literature	Representative paper
NM-Boost	imaginary boost deformation in integrable non-Hermitian systems	(Guo et al., 2023)
NM-Boost	boosted dark matter in non-minimal / multi-component dark sectors	(Agashe et al., 2014, Alhazmi et al., 2016, Fujiwara et al., 1 Jun 2026)
uBoost	boosting method for uniform selection efficiencies	(Stevens et al., 2013)
YIG/NM	NM = nonmagnetic-metal layer	(Kang et al., 2016)

The most explicit formal definition of NM-Boost is the one given in “Non-Hermitian boost deformation” (Guo et al., 2023). In that work, NM-Boost is not a machine-learning method and not a condensed-matter transport protocol; it is an analytic deformation principle for integrable Hamiltonians. The dark-matter papers use the same label in a different sense, namely as a shorthand for boosted dark matter in non-minimal sectors (Agashe et al., 2014, Alhazmi et al., 2016, Fujiwara et al., 1 Jun 2026).

2. Formal construction as an imaginary boost deformation

For ordinary integrable systems, the boost deformation is generated by the boost operator

$B[H(\kappa)] = \sum_x x\,h_x,$

and in the Hermitian case the deformation is

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$

NM-Boost is defined by removing the factor of $i$: $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)].$$ The deformation is “imaginary” because it is the analytic continuation of the real boost parameter to an imaginary one (Guo et al., 2023).

The paper places this construction in the same conceptual lineage as the Hatano–Nelson model. A nonreciprocal hopping induced by an imaginary gauge field can be understood as a boost-like transformation, and NM-Boost generalizes that viewpoint from simple free models to interacting integrable systems. In the conserved-charge language, the boost acts as

$[B,I_n]= i I_{n+1},$

and under the imaginary deformation the conserved charges satisfy

$$\frac{d I_n^\kappa(\lambda)}{d\kappa}=[B,I_n^\kappa(\lambda)]= i I_{n+1}^\kappa(\lambda), \qquad n>0.$$

The stated consequence is that the deformation complexifies the spectral parameter $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$0 of the transfer matrix and thus the model’s quasiparticle data. For quasiparticles, the momentum is shifted according to

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$1

3. Energy-twisted boundary conditions and Bethe-ansatz implementation

A central statement of the construction is that the same physics can be described either as a bulk imaginary boost deformation or as a boundary twist by the energy (Guo et al., 2023). For a chiral Dirac fermion on a ring, the energy-twisted boundary condition is written as

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$2

For a single-particle mode this becomes

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$3

which implies

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$4

The hallmark of NM-Boost is therefore an energy-dependent momentum quantization condition rather than a simple constant shift.

In coordinate Bethe ansatz language, the same structure appears as

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$5

This is the key Bethe-ansatz implementation: the quasiparticle momentum picks up an imaginary part proportional to its own energy. The paper also gives a geometric interpretation through the rescaled coordinate

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$6

for which periodicity in the new frame reproduces the same twisted condition. This suggests that NM-Boost generalizes the Hatano–Nelson or Galilean-transformation idea from a constant imaginary vector potential to an energy-dependent, nonlinear twist.

4. Free-fermion spectra, point-gap winding, and the skin effect

The continuum free-fermion example makes the spectral complexification explicit. For the quadratic dispersion

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$7

the energy-twisted condition gives

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$8

Combining these relations yields

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)]$$9

with two branches

$i$0

Thus even the simplest free model develops a two-branch complex spectrum (Guo et al., 2023).

Under periodic boundary conditions, the complex spectrum winds in the complex plane. The paper defines the winding number as

$i$1

The boosted spectrum can exhibit exceptional-point-like square-root singularities and point-gap topology. Under open boundary conditions, by contrast, the continuum model is reduced to the Hermitian free particle by the similarity transformation

$i$2

leading to

$i$3

The eigenstates are therefore exponentially localized at one edge: $i$4 gives left-edge localization and $i$5 gives right-edge localization. This is the non-Hermitian skin effect, with localization length

$i$6

The lattice free-fermion model exhibits the same mechanism in a nonlinear form. Its boosted dispersion satisfies

$i$7

For small $i$8, the spectrum forms an eight-shaped loop with winding number

$i$9

As $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$0 increases, the loops expand and eventually reconnect into cross-shaped structures.

5. Interacting integrable realizations

The Calogero–Sutherland model is used as an interacting integrable example in which the boosted single-particle energy obeys

$$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$1

The paper reports that, for both real and imaginary boosts, the spectrum is complexified; interactions change the density and organization of the spectral branches; the two-particle spectrum shows multiple loops or arcs depending on $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$2 and interaction strength; and with stronger interaction more states appear and the spectral clusters become denser, while the qualitative boost-induced complexification remains (Guo et al., 2023).

For the XXZ chain, the deformation modifies the Bethe equations to

$$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$3

In the two-particle sector, the spectrum reorganizes as the interaction strength is varied. The reported pattern is that weak interaction gives behavior similar to the free-fermion case, with multiple loops and loop breaking into arcs as $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$4 increases, whereas strong interaction produces extra clusters and an earlier transition from closed loops to open arcs. The onset of open curves occurs at smaller $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$5 as the interaction becomes stronger. The paper’s broader claim is that NM-Boost is not just a free-fermion trick: it survives in genuine interacting integrable models and can reveal nontrivial many-body restructuring of the complex spectrum.

6. Alternative usage in boosted dark matter

In dark-sector phenomenology, the same label is used for boosted dark matter in non-minimal or multi-component sectors. “(In)direct Detection of Boosted Dark Matter” introduces a two-component thermal relic model in which the dominant species $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$6 has no tree-level coupling to the Standard Model, the subdominant species $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$7 couples through a dark photon, and present-day annihilation

$$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$8

produces monoenergetic boosted $$\frac{dH(\kappa)}{d\kappa}=[B[H(\kappa)],H(\kappa)]$$9 particles with $B[H(\kappa)] = \sum_x x\,h_x,$0. The signal combines the production mechanism of indirect detection with the readout mechanism of direct detection, and for masses in the $B[H(\kappa)] = \sum_x x\,h_x,$1–$B[H(\kappa)] = \sum_x x\,h_x,$2 range the most promising signatures are electron scattering events pointing toward the Galactic center (Agashe et al., 2014).

“Boosted Dark Matter at the Deep Underground Neutrino Experiment” studies the same non-minimal logic for solar boosted dark matter. The model contains two stable dark-sector fermions, $B[H(\kappa)] = \sum_x x\,h_x,$3 and $B[H(\kappa)] = \sum_x x\,h_x,$4, with $B[H(\kappa)] = \sum_x x\,h_x,$5, a contact operator for $B[H(\kappa)] = \sum_x x\,h_x,$6, and a light dark photon mediator. The paper emphasizes DUNE’s liquid argon time projection chamber, using DUNE 10 kTon and DUNE 40 kTon, threshold energy $B[H(\kappa)] = \sum_x x\,h_x,$7 MeV, and angular resolution $B[H(\kappa)] = \sum_x x\,h_x,$8. Its central conclusion is that DUNE is exceptionally powerful for boosted dark matter from the Sun and that solar boosted dark matter can probe parameter space above $B[H(\kappa)] = \sum_x x\,h_x,$9 GeV (Alhazmi et al., 2016).

“Boosted dark matter via semi-annihilation in a radiative neutrino mass model” gives an explicit semi-annihilation realization. The model extends the two-loop radiative neutrino mass model of Ma by introducing a global $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$0 that is spontaneously broken by $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$1, leaving a remnant $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$2 symmetry. The lightest Dirac fermion $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$3 is the dark matter candidate and undergoes semi-annihilation,

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$4

while neutrino masses are generated at two loops through the same new particles and couplings. After imposing charged lepton flavor violation, electroweak precision, invisible-$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$5, Higgs-mixing, Big Bang nucleosynthesis, and potential-stability constraints, the paper finds that the mediator mass must be $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$6 to enhance the elastic scattering cross section with protons to $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$7, placing parts of the parameter space within reach of DUNE and DARWIN (Fujiwara et al., 1 Jun 2026).

7. Distinctions from adjacent terminology

NM-Boost should not be conflated with uBoost. uBoost is a modification of standard boosting designed for particle-physics analyses in which the goal is not only signal-background separation but also a uniform selection efficiency in a user-defined multivariate space. Its defining weight update is

$$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$8

where $$\frac{dH(\kappa)}{d\kappa}= i[B[H(\kappa)],H(\kappa)].$$9 is the usual misclassification weight and $i$0 enforces local efficiency uniformity for signal events. The method is implemented for boosted decision trees and is intended for amplitude analyses, Dalitz-plot analyses, and related cases in which optimizing a single integrated figure of merit is not the desired objective (Stevens et al., 2013).

Nor should the label be read through the unrelated abbreviation “NM” in YIG/NM multilayers, where NM means nonmagnetic-metal. In that setting the central claim is that spin current in Pt or Ta under ferromagnetic resonance is pumped not from the precessing YIG magnetization itself but from the magnetized NM surface produced by the magnetic proximity effect at the interface (Kang et al., 2016). This suggests that the meaning of NM-Boost is fixed almost entirely by disciplinary context. In integrable non-Hermitian physics it denotes a boost-based deformation framework that unifies spectral complexification, point-gap winding, and skin localization (Guo et al., 2023); in dark-sector phenomenology it denotes boosted dark matter from non-minimal sectors, where the distinctive signature is a relativistic dark particle produced by annihilation or semi-annihilation and detected through terrestrial scattering (Agashe et al., 2014, Alhazmi et al., 2016, Fujiwara et al., 1 Jun 2026).