Reversed Skin Dynamics Overview

• Reversed skin dynamics are phenomena where a boundary or 'skin' element reverses its expected accumulation direction, driven by momentum slices, phase shifts, or boundary conditions.
• In non-Hermitian lattices, controlled reversals occur without altering intrinsic couplings, challenging the conventional bulk-boundary correspondence with reciprocal mechanisms.
• Across fields, the concept explains delayed pathogen dominance in skin microbiome models and out-of-phase oscillations in nuclear skins, highlighting diverse reversal dynamics.

Searching arXiv for papers and terminology on “reversed skin dynamics” and related “skin effect” usage. “Reversed skin dynamics” is not a single standardized construct but a family of phenomena in which a “skin” degree of freedom evolves, accumulates, or oscillates opposite to a naive, short-time, or locally inferred direction. In non-Hermitian lattice physics, the phrase and closely related terminology refer to momentum-resolved, drive-controlled, or boundary-controlled reversals of skin-mode accumulation and information flow. In skin-microbiome population dynamics, it refers to long-time reversal from an apparently stable commensal-dominant regime to a pathogen-dominant stable state. In nuclear structure, it denotes pygmy dipole and quadrupole modes in which the neutron or proton skin oscillates out of phase with the core (Hofmann et al., 2019, Greugny et al., 2023, Tsoneva et al., 2012).

1. Terminological scope and recurring structure

Across the cited literature, the common element is not a shared formalism but a shared dynamical pattern: a boundary, surface, or skin subsystem exhibits behavior that reverses an initial expectation. In non-Hermitian systems, the expectation usually comes from local hopping asymmetry, reciprocity, or periodic-boundary spectra; in microbiome models, it comes from short-time observations near $40$–$50$ h; in nuclear response, it comes from distinguishing core motion from surface-layer motion.

Domain	“Skin” object	Reversal mechanism
Non-Hermitian lattices	Boundary-localized eigenmodes	Momentum, phase, or transverse-boundary control
Skin microbiome dynamics	Quasi-stable commensal-dominant state	Slow drift to a reversed stable state
Nuclear response	Neutron or proton skin	Out-of-phase oscillation against the core

A frequent misconception is that reversal requires reversing the microscopic couplings themselves. Several non-Hermitian constructions show the opposite: the direction of accumulation can be reversed by momentum slice, by the relative phase of periodic drives, or by boundary conditions in a transverse direction, without altering the original longitudinal asymmetric couplings (Hofmann et al., 2019, Wang, 21 May 2026, Yang et al., 2 Sep 2025). A second misconception is that short-time stationarity implies true stability. The microbiome model explicitly shows that a state that appears stable on a $2$-day window can be only quasi-stable on longer timescales (Greugny et al., 2023).

2. Reciprocal skin effect as the canonical non-Hermitian reversal

The clearest non-Hermitian realization is the reciprocal skin effect in a two-band $\pi$-flux model on the square lattice. The Hermitian part is

$$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$

to which one adds a purely imaginary diagonal hopping that preserves reciprocity but breaks Hermiticity,

$$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$

The resulting Bloch Hamiltonian satisfies $H(k_x,k_y)^T=H(-k_x,-k_y)$, and its periodic-boundary dispersion is $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$ (Hofmann et al., 2019).

The reversal appears after passing to a strip geometry with OBC in $x$ and PBC in $y$. For each fixed $50$0, the system becomes an effective one-dimensional non-Hermitian chain with asymmetric effective hoppings. The bulk decay length under OBC is

$50$1

Since $50$2 changes sign under $50$3, one has $50$4. Hence $50$5 localizes on the right boundary, $50$6 localizes on the left boundary, and $50$7 is delocalized because reciprocity is locally restored. The system is globally reciprocal, yet opposite longitudinal momenta exhibit opposite transverse accumulation (Hofmann et al., 2019).

This mechanism differs sharply from the standard one-dimensional non-Hermitian skin effect. In a Hatano–Nelson-type chain, non-reciprocal hopping $50$8 causes all modes to localize on the same boundary. In the reciprocal skin effect, reciprocity is not broken globally; instead, each fixed-$50$9 slice breaks reciprocity in an opposite way to its $2$0 partner. The corresponding periodic-boundary spectrum contains four exceptional points, and the mismatch between PBC and OBC spectra on the strip signals a breakdown of conventional bulk-boundary correspondence (Hofmann et al., 2019).

The effect was realized experimentally in a passive topolectrical circuit built from a $2$1 array of unit cells, with $2$2. By reconstructing the Green’s function $2$3, Fourier transforming to $2$4, and extracting admittance bands and eigenvectors, the experiment observed branch cuts ending in exceptional points under PBC and opposite edge localization for modes near $2$5 and $2$6 under OBC. Localization was quantified by the inverse participation ratio $2$7 (Hofmann et al., 2019).

3. Dynamical reversal in non-reciprocal and driven lattices

A distinct development concerns the dynamics of propagation against the preferred skin direction. In the non-Hermitian SSH chain,

$2$8

$2$9 localizes modes at the left boundary and $\pi$0 at the right. Directionality is quantified by quantum Liang information flow (QLIF),

$\pi$1

where the “frozen” evolution removes all Hamiltonian matrix elements touching the source site. The effective propagation velocity extracted from the onset time obeys

$\pi$2

so information propagates fastest with the skin direction and slowest against it. Three temporal regimes emerge: light-cone-bounded spreading, $\pi$3-dependent stabilization, and coherent oscillations. The asymmetry

$\pi$4

shows a “scissors effect,” approximately linear in $\pi$5 for $\pi$6, peaking at moderate skin localization and vanishing both for extreme localization and in the Hermitian-like limit (Yi, 20 Feb 2026).

Periodic driving introduces a different reversal channel. A static PT-symmetric two-band non-Hermitian chain has no skin effect because the exact PT symmetry forces the complex-energy spectrum to satisfy $\pi$7, precluding nonzero point-gap winding. With two time-periodic drivings

$\pi$8

the relative phase $\pi$9 enters the Floquet effective Hamiltonian through commutators proportional to $$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$0. For $$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$1, the commutator vanishes, PT symmetry is retained, and the winding

$$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$2

remains zero. For $$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$3, PT symmetry is broken, the PBC spectrum becomes a loop of nonzero area, $$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$4, and NHSE appears. The localization direction flips under $$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$5, since the effective $$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$6 and $$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$7 change sign with $$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$8 (Wang, 21 May 2026).

Reversal can also be induced non-locally, without modifying the directed longitudinal hoppings. In a class of non-Hermitian lattices, including a warm-up model

$$H_{\pi}(k_x,k_y)=t \begin{bmatrix} 2\cos k_y & 1+e^{-i k_x}\ 1+e^{i k_x} & -2\cos k_y \end{bmatrix},$$9

and higher-dimensional extensions such as

$$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$0

the periodic-boundary spectrum develops epicyclic loops whose winding chirality can reverse on subsegments. Under OBC, those segments generate left-skin modes even though the added hopping also points to the right. In the designed Kagome lattice, changing the transverse boundary parameter $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$1 switches the $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$2-accumulation from right-localized to left-localized while leaving the original longitudinal couplings unchanged. Wavepacket simulations show that under $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$3-OBC the packet can first drift right and then reverse left, ultimately settling near $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$4 while its norm grows as $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$5 (Yang et al., 2 Sep 2025).

A related mean-field bosonic realization occurs in a nonreciprocal two-leg ladder with synthetic magnetic flux. The Gross–Pitaevskii equations contain nonreciprocal intraleg hoppings $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$6, $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$7, inter-leg coupling $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$8, and interaction $$H(k_x,k_y)=H_{\pi}(k_x,k_y)- i\,r \begin{bmatrix} 0 & e^{i k_y}\ e^{-i k_y} & 0 \end{bmatrix}.$$9. The instantaneous average intraleg current

$H(k_x,k_y)^T=H(-k_x,-k_y)$0

distinguishes “chiral” motion, $H(k_x,k_y)^T=H(-k_x,-k_y)$1, from “antichiral” or reversed skin motion, $H(k_x,k_y)^T=H(-k_x,-k_y)$2. In the linear limit, the sign of each current is governed by $H(k_x,k_y)^T=H(-k_x,-k_y)$3, so $H(k_x,k_y)^T=H(-k_x,-k_y)$4 and $H(k_x,k_y)^T=H(-k_x,-k_y)$5 with the same sign produce antichiral motion on both legs. With interactions, the ladder supports skin-dominated, self-trapping, and trap-skin regimes (Chen et al., 2024).

4. Open-system and many-body extensions

Open-system formulations make reversal a property of steady states and relaxation spectra rather than only of eigenmode localization. A representative construction is a two-chain Lindbladian with reversed skin localization on the two legs. The density matrix evolves as

$H(k_x,k_y)^T=H(-k_x,-k_y)$6

with coherent ladder Hamiltonian

$H(k_x,k_y)^T=H(-k_x,-k_y)$7

and jump operators engineered so that chain $H(k_x,k_y)^T=H(-k_x,-k_y)$8 drives left while chain $H(k_x,k_y)^T=H(-k_x,-k_y)$9 drives right. The no-jump evolution is governed by an effective non-Hermitian Hamiltonian $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$0, whose asymmetric hoppings $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$1 generate opposite skin drives on the two chains (Feng et al., 2024).

The central dynamical observable is the Liouvillian gap,

$E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$2

with $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$3 the steady-state eigenvalue. Exact diagonalization in the one-particle sector shows a crossover from $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$4 at $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$5 to $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$6 as $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$7 increases. In the thermodynamic limit, even an arbitrarily small coupling produces a non-perturbative rearrangement of the Liouvillian spectrum and steady state, termed the critical Liouvillian skin effect. The many-body problem, treated by stochastic Schrödinger equation unraveling, exhibits a corresponding entanglement transition: for small $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$8, the steady-state entropy obeys an area law; for large $E_\pm(k)=\pm\sqrt{d_x^2+d_y^2+d_z^2}$9, it crosses to a logarithmic law, while mutual information and density-density correlations turn on and become long-ranged (Feng et al., 2024).

This result matters conceptually because it separates three layers of reversal. First, there is static reversed localization between the two chains. Second, there is dynamical delocalization of the skin steady state as $x$0 grows. Third, there is a thermodynamic singularity at $x$1, meaning that an apparently perturbative interchain coupling becomes non-perturbative in the $x$2 limit (Feng et al., 2024).

5. Reversed stability in skin microbiome population dynamics

In skin microbiome modeling, the relevant “skin” is literal rather than metaphorical, and the reversal concerns long-time ecological dominance. A three-variable ODE model tracks the apparent surface concentrations of commensal bacteria $x$3, opportunistic pathogens $x$4, and bacterially produced antimicrobial peptides $x$5. The full reduced system is

$x$6

The model starts from $x$7 parameters and, using steady-state constraints from published data, is reduced to $x$8 free parameters: $x$9 Calibration is performed by FO-LTL($y$0) specifications combined with global parameter optimization via CMA-ES, and sensitivity is studied by one-at-a-time and global analyses (Greugny et al., 2023).

On the $y$1–$y$2 h timescale, the model approaches a pseudo-steady state consistent with experiments. The pathogenic fit targets $y$3 at $y$4 h; a representative parameter set is $y$5, $y$6, $y$7, $y$8, $y$9. A healthy fit requires $50$00 at $50$01 h; a representative set is $50$02, $50$03, $50$04, $50$05. The pathogenic outcome is most affected by $50$06, while healthy balance is most sensitive to $50$07 and the carrying capacities. Initial conditions and $50$08 are relatively uncritical (Greugny et al., 2023).

The reversal appears only on longer timescales. Simulations out to roughly $50$09 h show that the apparent equilibrium reached around $50$10 days can be a quasi-stable branch

$50$11

followed by slow drift toward a true stable branch

$50$12

The switching time is estimated by

$50$13

where $50$14 is the slow exponential growth rate of $50$15 once $50$16. Thus an apparently healthy state can mask a hidden long-term tendency toward pathogen takeover (Greugny et al., 2023).

Tropical algebra clarifies why this quasi-stability is unusual. After rescaling in powers of a small $50$17, the model has one fast variable, $50$18, and two slow variables, $50$19 and $50$20. For the numerical example $50$21, tropical equilibration yields a quasi-stable branch with $50$22, $50$23, and a reversed stable branch with $50$24, $50$25. The slowness does not arise from a generic slow-fast separation but from a near-cancellation in the pathogen equation’s constant term,

$50$26

The paper therefore characterizes quasi-stability as non-generic, and generalizes the balance conditions to an $50$27-species adversarial model

$50$28

where reversal occurs once rare species leave the $50$29 regime and the system transitions to a new attractor (Greugny et al., 2023).

6. Nuclear skin oscillations and cross-field distinctions

In nuclear structure, “skin dynamics” refers neither to boundary accumulation nor to ecological stability, but to collective oscillations of the neutron or proton skin. Within a density-functional plus quasiparticle-phonon framework, the ground state is obtained from HFB equations with neutron and proton densities $50$30, $50$31 and skin thickness

$50$32

Small-amplitude excitations are built with QRPA phonons

$50$33

response functions

$50$34

and transition densities $50$35, with QPM multi-phonon admixtures used to fragment and shift the strength (Tsoneva et al., 2012).

The pygmy dipole resonance (PDR) and pygmy quadrupole resonance (PQR) are the relevant reversed skin modes. In Sn isotopes, the PDR is a cluster of $50$36 states at $50$37–$50$38. Their transition densities show proton and neutron motion in phase in the nuclear interior, but at the surface neutrons dominate with opposite sign, indicating that a thin neutron skin oscillates against an inert isospin-saturated core. The summed $50$39 strength below $50$40 grows monotonically with $50$41, for example from about $50$42 in $50$43 to about $50$44 in $50$45, while $50$46 grows from $50$47 to $50$48. The QPM shifts about $50$49–$50$50 of the strength downward, roughly doubling the QRPA-only $50$51 in the $50$52 window (Tsoneva et al., 2012).

An analogous pattern appears in the quadrupole channel. Low-lying $50$53 states at $50$54–$50$55 display surface neutron oscillations out of phase with the core, and the summed $50$56 for these PQR states rises with skin thickness. In proton-rich $50$57Sn–$50$58Sn, the mode reverses character and a proton skin drives the PQR. In this context, “reversed” refers to phase opposition between skin and core rather than to left-right transport (Tsoneva et al., 2012).

The cross-field distinction is essential. In reciprocal or non-reciprocal non-Hermitian lattices, reversed skin dynamics concerns spectral winding, localization length, information-flow asymmetry, or boundary-controlled amplification. In the skin microbiome model, it concerns non-generic quasi-stability and delayed dominance reversal. In nuclear physics, it concerns out-of-phase oscillation of a surface layer. The shared terminology is therefore descriptive rather than unificatory: the cited works use “skin” for different physical objects, and the “reversal” is defined relative to different baselines in each domain.