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Sandbox Physics

NON-HERMITIAN SKIN EFFECT

A Cut That Moves Every Mode

Close a 41-site loop, cut one link, and watch its right eigenmodes concentrate at a boundary. Reverse the hopping bias, inspect the complex spectrum, or release a central wave packet.

Interactive modelA Cut That Moves Every Mode
THE EDGE IS PART OF THE PHYSICS

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.

EVERY OUTCOME COUNTS

An edge full of modes. A record full of losses.

All forty-one right modes

Each row is a separately normalized right eigenmode, not a simultaneous particle population. Brightness is square-root probability, fixed from zero to 0.7. Selecting a row prepares that mode anew.

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. 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.

Ideal loss-or-position readout, with no detector inefficiency. Time and display-scale changes preserve the record. A new preparation clears it. 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 establish the physical context. This Lab uses a stated linear model and illustrative parameters, not a reproduction of their experimental data.

Physics tutorial

A Cut That Moves Every Mode

BackgroundClose a 41-site loop, cut one link, and watch its right eigenmodes concentrate at a boundary. Reverse the hopping bias, inspect the complex spectrum, or release a central wave packet.

Why it mattersLearn to distinguish a changing wave shape from the probability that anything remains.

Start with the essentials

Focus question
Open one connection. The modes crowd an edge.
One-sentence intuition
Asymmetric hopping and an open boundary turn extended right eigenmodes into exponential edge profiles. A survival-conditioned image alone cannot reveal how many trials were lost.

Core mathematical model

Generator

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 inverse hopping units. The propagation is not renormalized.

Open-chain right modes

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

Here site indices run from zero to forty. Reversing the bias reverses the exponential envelope.

Common difficulties

Shape is not survival

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

Better mental modelThe display scale and conditioning are stated explicitly. The dynamics is passive; total survival never grows.

Model scope

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

Better mental modelA passive, uniformly loss-shifted Hatano–Nelson model of one excitation. Bright conditional shapes can coexist with very small survival. Changing controls prepares a new experiment; it is not a mid-run boundary quench or a many-body skin measurement.

Run the experiment

  1. 01

    Make a contrast

    Compare the first two presets at the same time.

    What to observe: Every control change prepares a new state with the selected boundary. Cutting the link is not applied during this run.
  2. 02

    Inspect the probability

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

    What to observe: The conditional shape sums to one even when very few trials survive.
  3. 03

    Keep every outcome

    Acquire 1,024 independent preparations. Export lost outcomes as well as measured sites.

    What to observe: A bright conditional image does not increase the number of detected events. The snapshot stays frozen when time moves.
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