---
title: "A Cut That Moves Every Mode Interactive Lab | Sandbox Physics"
description: "Open one connection. The modes crowd an edge. 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."
canonical_url: "https://sandboxphysics.com/labs/non-hermitian-skin-effect/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/non-hermitian-skin-effect/"
last_modified: "2026-10-22"
---

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 simulation: open this page in a web browser to change the parameters and run the model.

A 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.

THE EDGE IS PART OF THE PHYSICS

CONDITIONAL WEIGHT IN FIVE EDGE SITES

$0\%$

EVOLUTION TIME

$\tau=0$

Complex phase $-\pi$ $\pi$

Actual survival $100\%$

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.

0 0.7

### 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 RATIO $0$

MAX LOSS-BALANCE ERROR $0$

### 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

$$
\tau=Jt/\hbar,\qquad E_*=J
$$

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.

$$
\begin{gathered}i\dot\psi_n=-e^g\psi_{n-1}-e^{-g}\psi_{n+1}\\-i\Gamma\psi_n,\quad\Gamma=2\sinh|g|\end{gathered}
$$

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.

$$
\begin{gathered}E_m^{\rm open}=-2\cos\!\frac{m\pi}{42}-i\Gamma\\E_k^{\rm ring}=-2\cosh g\cos k\\+2i\sinh g\sin k-i\Gamma\end{gathered}
$$

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.

$$
\begin{gathered}S=\sum_n|\psi_n|^2,\quad P_{\rm lost}=1-S\\p_n=|\psi_n|^2,\quad p_{n|\rm survive}=p_n/S\end{gathered}
$$

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.

- [Roccati — Non-Hermitian skin effect as an impurity problem (2021)](https://arxiv.org/abs/2105.01197)
- [Shen et al. — Non-Hermitian skin effect on a digital quantum computer (2025)](https://www.nature.com/articles/s41467-025-55953-4)

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

**Background** 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.

**Why it matters** Learn 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

$$
i\dot\psi_n=-e^g\psi_{n-1}-e^{-g}\psi_{n+1}-i\Gamma\psi_n
$$

Time is measured in inverse hopping units. The propagation is not renormalized.

#### Open-chain right modes

$$
u_n^{(m)}\propto e^{gn}\sin\!\frac{m\pi(n+1)}{N+1}
$$

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

### Common difficulties

#### Shape is not survival

**Typical misconception** The wave gets brighter, so probability must have been created.

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

#### Model scope

**Typical misconception** This is a calibrated apparatus or an interacting many-body experiment.

**Better mental model** A 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.

Physical conclusion

A 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.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)