Exotic critical states as fractional Fermi seas in the one-dimensional Bose gas

この論文は、反発から引力へと相互作用を周期的に変化させることで、一次元ボース気体が従来のトムナガ・ルッティンガー液体とは異なる臨界相である「分数フェルミ海」を形成し、その性質を一般化流体力学を用いて予測することを示しています。

要約

1. 舞台設定：極低温の「量子の海」

まず、この実験の舞台は、極寒の空間に閉じ込められた**「ボース原子ガス」**です。
通常、原子は互いに反発したり、くっついたりしますが、この実験では「1 次元（細い管の中）」に閉じ込め、温度を絶対零度近くまで下げています。

• いつもの海（トムナガ・ルッティガー液体）：
  普通の量子の世界では、原子たちは「フェルミの海」という状態になります。これは、**「満員電車」**に例えられます。一人の席（エネルギー状態）には最大 1 人しか乗れません（パウリの排他原理）。この状態では、電車の動きが規則正しく、波のような振る舞い（相関）を見せます。

• 今回の発見（分数のフェルミの海）：
  この研究では、**「満員電車」ではなく、「空席だらけの電車」を作り出しました。
  通常、原子は「1 人しか乗れない席」ですが、この実験では「2 人、3 人、あるいはもっと多くの人が乗れる席」のように見えます。しかし、実際には「1 人しか乗れない」のに、「乗っている人が 1 人だけ」という奇妙な状態（分数の占有）が実現されたのです。
  これを「分数のフェルミの海（Fractional Fermi Sea）」**と呼んでいます。

2. 魔法の儀式：相互作用の「サイクル」

では、どうやってこの奇妙な「空席だらけの海」を作ったのでしょうか？
答えは、**「原子同士の反発力を、リズミカルに変える魔法の儀式」**です。

1. スタート： 原子同士が少し反発し合う状態から始めます。
2. 変化： 反発力を強めて、原子同士が「絶対に触れ合わない（硬い球）」状態にします。
3. ジャンプ： さらに勢いをつけて、反発から**「強烈な引力」**へと一瞬で切り替えます（ここが重要！）。
4. 戻り： 再び反発力をゼロにして、元に戻ります。

この「反発→引力→戻り」という動きを、**「回転（サイクル）」と呼びます。
この儀式を「W 回」**繰り返すと、不思議なことが起きます。

• アナロジー：
  Imagine a crowded dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.
  Imagine you have a dance floor where everyone is trying to avoid bumping into each other.

技術概要

この論文「Exotic critical states as fractional Fermi seas in the one-dimensional Bose gas（一次元ボース気体における分数フェルミ海としての異種臨界状態）」の技術的な要約を以下に記します。

1. 研究の背景と課題

• 背景: 低温における一次元量子多体系は、一般的にトモナガ・ラッティンガー液体（TLL）理論によって記述されます。TLL は、有効的なフェルミ海の存在に基づき、べき乗則に従う相関関数やフリーデル振動（Friedel oscillations）を示すことが知られています。
• 課題: 従来の TLL 理論を超えた、新しい普遍性クラス（ユニバーサリティクラス）を持つ臨界状態の存在が示唆されています。特に、ハルダンの提案した「一般化排除統計（GES）」に基づく分数フェルミ海（Fractional Fermi Sea: FFS）の概念は、フェルミオンとボースンの中間的な統計を持つ系を記述しますが、通常は基底状態や特定の統計的制約下でのみ議論されてきました。
• 目的: 本論文では、平衡状態ではなく、非平衡過程を通じて、相互作用するボース気体（Lieb-Liniger モデル）において、分数フェルミ海（FFS）が実現されることを理論的に示し、その相関関数が従来の TLL 理論とは異なる「異種臨界性（Exotic criticality）」を示すことを明らかにすることを目的としています。

2. 手法と理論的枠組み

• モデル系: 一次元接触相互作用を持つ Lieb-Liniger (LL) モデル。
• 非平衡プロトコル: 相互作用結合定数 $g_{1D}$ を時間的にゆっくりと周期的に変化させる「相互作用サイクル」を提案します。
  • 弱反発 ($g_{1D} > 0$) $\to$ 強反発 $\to$ トンクス・ギラールドー（TG）領域 $\to$ 無限大の引力 $\to$ 無限大の引力から TG-超トンクス・ギラールドー（sTG）遷移を通過 $\to$ 無限大の引力 $\to$ 非相互作用点 ($g_{1D}=0$) $\to$ 再び反発領域へ戻る。
  • このサイクルを $W$ 回繰り返します。
• 理論的手法:
  • 一般化流体力学（Generalized Hydrodynamics: GHD）: 可積分モデルの非平衡ダイナミクスを記述する枠組み。LL モデルには無限個の保存量があり、これらに基づく一般化ギブスアンサンブル（GGE）を用いて状態を記述します。
  • モンテカルロ法: 熱力学極限における運動量分布および一粒子相関関数 $g^{(1)}(x)$ を数値的に計算するために、ベテ・アンサツ（Bethe ansatz）の行列式表現とモンテカルロ法を組み合わせました。
• 量子アディアバティック性（QA）との比較: 従来の QA 的なアプローチ（ホロノミーサイクル）との整合性を確認しつつ、GHD の枠組みが熱力学極限や可積分性の破れに対してより頑健であることを示しました。

3. 主要な結果

• 分数フェルミ海（FFS）の実現:
  • 相互作用サイクルを $W$ 回繰り返すことで、初期の基底状態（最大占有率 1 のフェルミ海）が、最大占有率が $\vartheta(\lambda) \leq 1/(2W+1)$ に制限された新しい GGE 状態へと写像されることが示されました。
  • これは、有効的な排除統計パラメータ $\alpha = 2W+1$ を持つ分数フェルミ海（FFS）の形成に対応します。ただし、これは統計的制約によるものではなく、可積分系の保存則に基づく非平衡投影として実現されます。
• 異種臨界性の発見:
  • 一粒子相関関数 $g^{(1)}(x)$ の解析により、FFS 状態は従来の TLL 理論とは異なる特徴を持つことが明らかになりました。
    • フリーデル振動（FO）: 任意の反発相互作用強度において、明確なフリーデル振動が観測されます。その振動数は密度 $n$ の $2\pi W$ 倍に比例し、サイクル回数 $W$ に依存します。
    • 二重のべき乗則減衰: 相関関数は、短距離と長距離で異なるべき乗則（power-law）で減衰します。両者の遷移点（クロスオーバー距離 $\bar{x}$）は相互作用強度に依存して変化します。
    • TLL 理論との乖離: 従来の TLL では、低エネルギー励起はフェルミ端のみに由来しますが、FFS 状態では占有率が最大ではないため、バルク（内部）のモードも低エネルギー励起に関与し、これが異種臨界性の源となっています。
• 可逆性の破れ:
  • サイクルを逆方向にたどる際、$g_{1D}=0$ を通過する際に束縛状態（Bethe strings）が形成され、エントロピーが急激に増加します。これにより、サイクルの可逆性が破れ、「階段状のエントロピー増加」が生じることが示されました。

4. 貢献と意義

• 理論的貢献:
  • 非平衡過程を通じて、可積分系において「分数フェルミ海」が実現されることを初めて示しました。
  • 従来の TLL 理論の枠組みを超えた、新しい臨界相（Exotic critical phase）の存在とその物理的性質（振動するべき乗則減衰など）を詳細に記述しました。
  • GHD と QA の関係を明確にし、非平衡可積分系のダイナミクス理解を深めました。
• 実験的意義:
  • この理論的予測は、極低温原子気体（特にセシウム原子）を用いた実験と直接対応しています。共同投稿された実験論文 [12] と合わせて、GHD による予測が実験的に検証され、フリーデル振動が観測されたことが報告されています。
  • 可積分性の破れ（双極子相互作用など）が FFS の安定性に与える影響についても言及しており、将来の実験系設計への指針となります。

5. 結論

本論文は、相互作用サイクルを駆使して一次元ボース気体を非平衡状態に導くことで、分数フェルミ海という新しい量子状態を創出できることを示しました。この状態は、従来のトモナガ・ラッティンガー液体理論では説明できない特異な臨界振動（振動するべき乗則相関）を示し、量子多体系における非平衡臨界現象の新たなパラダイムを提供するものです。