Geometry-Protected Pure Decay Modes

• The paper demonstrates that geometry protection enforces a smooth exponential decay profile in systems like directed graphs, Kitaev chains, and cylindrical plasmas.
• It utilizes gauge engineering and non-reciprocal hopping alongside Lindblad dynamics to precisely control decay rates and mode isolation.
• The findings have practical implications for single-mode lasers, quantum sensors, and state transfer, showing that protection can coexist with measurable decay.

Searching arXiv for the cited papers and closely related work to ground the article. arXiv search: geometry-protected pure decay modes, Majorana edge dissipation, cylindrical plasma topological damping. In directed-graph networks, geometry-protected pure decay modes are eigenstates exhibiting smooth exponential amplitude decay along directed paths (Liu et al., 15 May 2026). In dissipative Kitaev chains, the same expression is used for Majorana qubits whose decay rates are exponentially small in the number of protected edge sites (Carmele et al., 2015). In cylindrical magnetized plasmas, collisionlessly damped yet topologically protected surface plasma waves realize an analogous combination of protection and decay (Rajawat et al., 1 Aug 2025). Across these settings, the protected feature is not the absence of dissipation; rather, it is the persistence of a constrained spatial profile, boundary localization, or unidirectional propagation while the mode undergoes a well-defined decay process.

1. Directed-graph realization

The directed-graph construction begins from an $N$-site network with binary adjacency matrix $a_{ji}\in\{0,1\}$, where $a_{ji}=1$ denotes a directed edge from site $i$ to site $j$. With non-reciprocal hopping amplitudes $t_{\rm r}$ and $t_{\rm l}$, the Hamiltonian is

$$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$

All on-site energies are set to zero, or to a common constant. A synthetic gauge field targeting mode index $k\in\{0,\dots,N-1\}$ is introduced by phase-compensated non-reciprocal hopping,

$$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$

with $a_{ji}\in\{0,1\}$0 (Liu et al., 15 May 2026).

A directed graph satisfying the mirror-type connectivity

$a_{ji}\in\{0,1\}$1

admits a complete basis of eigenmodes whose eigenvalues lie strictly on the imaginary axis. The mode ansatz is

$a_{ji}\in\{0,1\}$2

and the key amplitude property is

$a_{ji}\in\{0,1\}$3

which yields monotonic exponential decay along the direction of the arrows. In the fully connected case, a single dominant mode naturally emerges with a large, tunable energy gap from the rest. The corresponding pure-decay gap is

$a_{ji}\in\{0,1\}$4

and it grows monotonically with system size $a_{ji}\in\{0,1\}$5.

The defining feature of this construction is that the exponential amplitude profile is enforced by directed connectivity and $a_{ji}\in\{0,1\}$6, rather than by on-site gain and loss. Because there are no on-site gains or losses and all non-zero matrix elements lie off the diagonal, suitably chosen phases of $a_{ji}\in\{0,1\}$7 yield a purely imaginary spectrum. The paper explicitly contrasts this with gain–loss balancing and exceptional-point engineering.

2. Geometry protection, gauge selection, and higher-dimensional extensions

Without gauge, the mode with $a_{ji}\in\{0,1\}$8 maximizes the imaginary part of $a_{ji}\in\{0,1\}$9 and decays slowest. Gauge engineering shifts $a_{ji}=1$0, so the condition for the dominant mode becomes

$a_{ji}=1$1

This promotes any desired pure decay mode to the dominant position while preserving its amplitude profile (Liu et al., 15 May 2026).

The same framework extends beyond single-mode selection. In bipartite half-connected graphs, obtained by setting $a_{ji}=1$2 only when $a_{ji}=1$3, two modes, $a_{ji}=1$4 and $a_{ji}=1$5, maximize $a_{ji}=1$6 and form a well-isolated pair. Gauge tuning then allows selection of any desired pair among the $a_{ji}=1$7 pure-decay modes. In higher dimensions, a tensor-product Hamiltonian

$a_{ji}=1$8

produces factorized eigenmodes and additive eigenvalues $a_{ji}=1$9. One direction can supply the imaginary spectrum while orthogonal directions supply real frequency splitting, generating modes with the same decay rate but separated in real frequency.

Several general bounds are given. For an $i$0-site fully-connected directed graph with $i$1, the pure-decay gap scales as

$i$2

for large $i$3. Potential applications identified in the paper are single-mode lasers, sensors, and quantum processing and state transfer. The common mechanism is mode isolation through geometry and non-reciprocity rather than through spatial defects, $i$4-symmetry, or exceptional points.

3. Dissipative Kitaev chains and Majorana pure-decay modes

In the Kitaev-chain setting, the starting point is the standard Hamiltonian for spinless fermions $i$5 on a one-dimensional lattice of length $i$6,

$i$7

With Majorana operators

$i$8

the topological phase $i$9 for $j$0 supports two zero-energy Majorana modes, $j$1 and $j$2, which remain decoupled in the ideal limit $j$3, $j$4 (Carmele et al., 2015).

Coupling to a Markovian bath that removes or injects single fermions is modeled by the Lindblad master equation

$j$5

For pure loss, $j$6 and $j$7; for pure gain, $j$8 and $j$9. In many of the numerics, only bulk sites $t_{\rm r}$0 are coupled to the bath, leaving $t_{\rm r}$1 protected sites at each edge.

A convenient edge-mode observable is

$t_{\rm r}$2

with autocorrelation

$t_{\rm r}$3

Starting from an initial density $t_{\rm r}$4 with $t_{\rm r}$5, $t_{\rm r}$6 quantifies decoherence of the edge qubit. In the clean chain with symmetry-breaking loss, the long-time decay is purely exponential,

$t_{\rm r}$7

with $t_{\rm r}$8, where $t_{\rm r}$9 is the number of protected sites and $t_{\rm l}$0 is the localization length of the Majorana wave function. The data therefore show that even a small protected region exponentially slows the decay.

4. Disorder, many-body localization, and the stretched-exponential regime

When strong bond or site disorder drives the closed chain into an Anderson- or many-body-localized regime, the decay of the edge autocorrelation crosses over from exponential to stretched exponential. For pure-loss dynamics,

$t_{\rm l}$1

The paper reports that this behavior is seen both in the noninteracting limit and in the interacting MBL case, and that the exponent is universal, within numerical accuracy (Carmele et al., 2015).

The proposed origin of the $t_{\rm l}$2 law is heuristic rather than fully rigorous. The argument is that local excitations created by loss events remain localized except for rare Griffiths regions where they can hop anomalously far. The survival amplitude of the edge mode is then dominated by the slowest-decaying rare-region processes, producing a broad distribution of local decay rates. Assuming

$t_{\rm l}$3

down to small $t_{\rm l}$4, one obtains

$t_{\rm l}$5

Numerically, $t_{\rm l}$6 is extracted.

The numerical evidence is split by dynamical regime. For $t_{\rm l}$7, the quadratic Lindblad problem is solved up to $t_{\rm l}$8 sites by Prosen’s covariance-matrix method. For $t_{\rm l}$9, the MBL regime is treated by TEBD/MPO techniques for up to $$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$0 sites. In the clean chain, $$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$1 versus $$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$2 is linear; in the disordered chain, $$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$3 versus $$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$4 is linear. A common misconception is that topological edge protection in this setting implies non-decay. The results show the opposite: symmetry-breaking loss or gain does destroy the exact conservation of the Majorana qubit, but geometry and disorder strongly suppress the decay and change its functional form.

5. Cylindrical plasma edge modes and continuum damping

In cylindrical magnetized plasmas, the relevant objects are topologically protected surface plasma waves at a plasma–vacuum boundary. The plasma occupies $$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$5, vacuum lies at $$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$6, and the cold-plasma Maxwell equations are Fourier transformed as $$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$7. The dielectric tensor is

$$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$8

with

$$H \;=\;\sum_{i\neq j}\Bigl(t_{\rm r}\,a_{j i}\,\hat c_j^\dagger\hat c_i \;+\;t_{\rm l}\,a_{i j}\,\hat c_i^\dagger\hat c_j\Bigr).$$9

and $k\in\{0,\dots,N-1\}$0 as given in the model. Eliminating transverse fields yields coupled equations for $k\in\{0,\dots,N-1\}$1 and $k\in\{0,\dots,N-1\}$2, reducible to

$k\in\{0,\dots,N-1\}$3

with regular plasma solutions in terms of $k\in\{0,\dots,N-1\}$4 and vacuum solutions in terms of $k\in\{0,\dots,N-1\}$5. Continuity of $k\in\{0,\dots,N-1\}$6 at $k\in\{0,\dots,N-1\}$7, or integration of the four-component shooting system across a smooth transition layer, yields the dispersion determinant $k\in\{0,\dots,N-1\}$8 (Rajawat et al., 1 Aug 2025).

The crucial distinction is between a hard-plasma interface and a smooth plasma interface. The hard interface yields an undamped edge-mode curve $k\in\{0,\dots,N-1\}$9 that flattens unphysically at large $$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$0. With a smooth interface of width $$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$1, the branch for $$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$2 is strictly real and undamped, whereas for $$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$3 one obtains a complex-frequency quasi-mode $$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$4. The damping mechanism is collisionless and arises from resonant coupling to a continuum of upper-hybrid modes localized within the smooth transition layer.

Above the local upper-hybrid frequency, the factor $$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$5 becomes singular at $$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$6 defined by

$$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$7

Analytic continuation of the dispersion determinant into the lower-half $$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$8-plane yields an imaginary contribution from the residue, and the temporal decay rate is

$$H_{ij}^{(k)} =t_{\rm r}\,a_{j i}\,e^{-\,i\,k\,\varphi\,|j-i|},\qquad H_{ji}^{(k)} =t_{\rm l}\,a_{i j}\,e^{+\,i\,k\,\varphi\,|j-i|},$$9

Similarly, solving $a_{ji}\in\{0,1\}$00 gives the spatial attenuation constant

$a_{ji}\in\{0,1\}$01

For a linear density ramp, both $a_{ji}\in\{0,1\}$02 and $a_{ji}\in\{0,1\}$03 scale proportionally to $a_{ji}\in\{0,1\}$04.

6. Protection without immunity to damping

The three realizations can be compared directly:

Setting	Protected object	Decay law or mechanism
Directed graph	Eigenmode with monotonic amplitude profile	Purely imaginary eigenvalue, exponential spatial decay
Kitaev chain	Majorana edge qubit	Exponential in clean systems; stretched exponential with $a_{ji}\in\{0,1\}$05 under disorder
Cylindrical plasma	Unidirectional TSPW	Temporal and spatial damping from upper-hybrid continuum coupling

A central point is that protection and decay are not contradictory. In the directed-graph problem, geometry protection means that the smooth exponentially decaying eigenstate survives without fine-tuned gain–loss balance and without exceptional-point physics (Liu et al., 15 May 2026). In the Kitaev-chain problem, topology localizes the qubit at the boundary, protected edge sites make the decay rate exponentially small, and strong disorder or MBL converts the decay into a stretched exponential (Carmele et al., 2015). In the plasma problem, the nonzero Chern number of the bulk band gap guarantees unidirectionality and absence of back-reflection even though the edge mode loses energy collisionlessly to a continuum of upper-hybrid resonances (Rajawat et al., 1 Aug 2025).

Two recurrent misconceptions are explicitly ruled out by these results. First, protection does not imply zero damping: the protected feature may instead be an amplitude profile, a boundary localization pattern, or immunity to backscattering. Second, pure-decay behavior does not require balanced gain and loss. In directed graphs it is enforced by off-diagonal non-reciprocal connectivity; in plasmas it follows from causal analytic continuation through a continuum resonance; in dissipative Kitaev chains it emerges from the spatial separation of edge support from lossy bulk regions and is further enhanced by localization.

Taken together, these works suggest a broad research program in which geometry, topology, and non-reciprocity are used to engineer modes that remain structurally identifiable under open-system evolution. The implementations differ sharply in microscopic mechanism, but they share a common operational theme: decay is shaped, isolated, and in some cases dramatically slowed by the spatial organization of the system rather than eliminated altogether.