Skip to main content
Sandbox Physics

Non-Hermitian skin effect

A Cut That Moves Every Mode

Why does cutting one link in a ring where hopping favors one direction push every mode to one edge? Because the cut gives the chain two ends, and with ends every mode becomes a standing wave multiplied by a factor that grows exponentially toward the end that hopping favors, so all of them pile up at that edge. On the closed ring, every mode is instead a wave running evenly around the loop. Reversing the bias sends the modes to the other end.

Close a ring of 41 sites, cut one link, and watch the right eigenmodes pile up at one edge. Reverse the hopping bias, look at the complex spectrum, or release a wave packet from the center.

Interactive modelA Cut That Moves Every Mode
How the boundary changes the wave

CONDITIONAL WEIGHT IN FIVE EDGE SITES

0%0\%

EVOLUTION TIME

τ=0\tau=0
Complex phase−π-\piπ\pi

Actual survival100%100\%

The arc is a schematic layout of forty-one sites. Only the end link changes. Height is square-root probability; crest hue is phase. Controls prepare a new state, rather than opening a link mid-run.

Count every outcome

Modes crowd the edge, yet the record can be almost all losses.

All forty-one right modes

Each row is a separately normalized right eigenmode; the rows are not particles present at the same time. Brightness is the square root of probability, on a fixed scale from zero to 0.7. Selecting a row prepares that mode again.

00.7

A spectrum changes with its boundary

Cyan ellipse: closed-loop energies. Amber segment: open-chain energies. The highlighted point belongs to the selected boundary and mode. The imaginary axis includes the passive loss shift.

HOPPING RATIO00
MAX LOSS-BALANCE ERROR00

The probability at this instant

Independent simulated preparations

Count the survivors and keep the losses.

Each of 1,024 fresh preparations produces one outcome: lost, or detected at a site. The event grid uses amber for loss and cyan for survival. The histogram counts only those recorded surviving sites.

The readout ideally reports either a loss or a site, with no detector inefficiency. Changing the time or the display scale keeps the record; a new preparation clears it. The export includes every loss and every site, plus a separately labelled model reference.

Equations, loss and model boundaries
τ=Jt/ℏ,E∗=J\tau=Jt/\hbar,\qquad E_*=J

The hopping energy sets the units. Site number increases from left to right in the data plots. The three-dimensional layout is schematic; no physical length or calibrated apparatus is assigned.

iψ˙n=−egψn−1−e−gψn+1−iΓψn,Γ=2sinh⁡∣g∣\begin{gathered}i\dot\psi_n=-e^g\psi_{n-1}-e^{-g}\psi_{n+1}\\-i\Gamma\psi_n,\quad\Gamma=2\sinh|g|\end{gathered}

The directional couplings require an effective open-system construction. With the stated uniform shift, the anti-Hermitian loss matrix is positive semidefinite, so an absorbing-vacuum loss-channel realization exists. This does not model independent local decay at every site or a particular experimental circuit.

Emopen=−2cos⁡ ⁣mπ42−iΓEkring=−2cosh⁡gcos⁡k+2isinh⁡gsin⁡k−iΓ\begin{gathered}E_m^{\rm open}=-2\cos\!\frac{m\pi}{42}-i\Gamma\\E_k^{\rm ring}=-2\cosh g\cos k\\+2i\sinh g\sin k-i\Gamma\end{gathered}

Open modes carry an exponential envelope multiplying a standing wave; ring modes are plane waves. The index labels each spectrum separately, rather than identifying a mode followed adiabatically through a boundary change. The displayed map is the right-mode probability, not a biorthogonal density.

S=∑n∣ψn∣2,Plost=1−Spn=∣ψn∣2,pn∣survive=pn/S\begin{gathered}S=\sum_n|\psi_n|^2,\quad P_{\rm lost}=1-S\\p_n=|\psi_n|^2,\quad p_{n|\rm survive}=p_n/S\end{gathered}

The propagator uses fourth-order Runge–Kutta with a step no larger than 0.01. It never renormalizes the evolving state. A separately integrated loss rate checks the probability balance. The bright conditional display divides by survival; raw acquisition always includes lost trials. No gain, many-body statistics or continuously observed particle paths are claimed.

These sources give the physical background. This Lab uses the linear model described here with illustrative parameters; it does not reproduce their experimental data.

Physics tutorial

A Cut That Moves Every Mode

BackgroundClose a ring of 41 sites, cut one link, and watch the right eigenmodes pile up at one edge. Reverse the hopping bias, look at the complex spectrum, or release a wave packet from the center.

Why it mattersLearn to tell the shape of a wave apart from the probability that anything is left.

Start with the essentials

Focus question
Why does cutting one link in a ring where hopping favors one direction push every mode to one edge?
One-sentence intuition
With asymmetric hopping and an open boundary, the right eigenmodes stop spreading over the whole chain and pile up at an edge with an exponential profile. An image conditioned on survival cannot tell you how many trials were lost.

Core mathematical model

Equation of motion

iψ˙n=−egψn−1−e−gψn+1−iΓψni\dot\psi_n=-e^g\psi_{n-1}-e^{-g}\psi_{n+1}-i\Gamma\psi_n

Time is measured in units of the inverse hopping. The state is not renormalized as it evolves.

Right eigenmodes of the open chain

un(m)∝egnsin⁡ ⁣mπ(n+1)N+1u_n^{(m)}\propto e^{gn}\sin\!\frac{m\pi(n+1)}{N+1}

In this formula the sites are numbered from zero to forty. Reversing the bias flips the exponential envelope to the other edge.

Common difficulties

A bright shape does not mean high survival

Typical misconceptionThe wave gets brighter, so probability must have been created.

Better mental modelThe display scale and whether the image is conditioned on survival are both labeled. The dynamics is passive, so the total survival probability never grows.

What the model covers

Typical misconceptionThis simulates a calibrated apparatus or an interacting many-body experiment.

Better mental modelThis is a passive Hatano–Nelson model of one excitation, shifted by a uniform loss. A conditional shape can look bright even when the survival probability is very small. Changing a control prepares a new experiment; it does not cut the ring in the middle of a run, and it is not a many-body skin measurement.

Run the experiment

  1. 01

    Compare two presets

    Compare the first two presets at the same time.

    What to observe: Each change of a control prepares a new state with the chosen boundary. The link is not cut partway through a run.
  2. 02

    Check how much probability remains

    Switch between all trials and surviving trials. Keep an eye on the survival meter.

    What to observe: The conditional shape always adds up to one, even when very few trials survive.
  3. 03

    Keep every outcome

    Acquire 1,024 independent preparations, and export the lost outcomes along with the measured sites.

    What to observe: A bright conditional image does not mean more events were detected. The sampled snapshot stays fixed when you move the time.
FeedbackSomething broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy