diagram.

**Case II:** The second line annihilates two rows of length $r$ and $k$ (k&gt;r) and generates two new rows of length $k-n$ and $r+n$, which makes long row shorter and short row longer, but not longer than the new shorter row, also, it squeezes the Young diagram.

**Case III:** The third line annihilates one row of length $k$ and one column $r$ and generates a shorter row (length $r-n$) and a longer column (length $k+n$).

**Case IV:** The fourth line annihilates two column of length $r$ and $k$ (k&gt;r) and one row of length $s$ and generates a single column of length $r+k+s$ .

**Case V:** The fifth line annihilates a long row of length $r+k+s$ into three short columns of lengths $r$, $k$, $s$ respectively .

From the above analysis, we conclude that the $H_{t}$, when acting on a Schur state, will generate series of squeezed states. We call this property the fermionic triangularization.

Integrability
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The triangular property means the eigenstate can be understood as $|P_{\lambda}(\psi\psi^{*})\rangle$, with $$P_{\lambda}=s_{\lambda}+\sum_{\mu<\lambda}c_{\lambda,\mu}s_{\mu}\,.$$ This ansatz is due to the eigenstate of bosonic $H$ is a function of bosonic oscillators $a_{-n}$’s. When $a_{-n}$ acts on a coherent basis, the eigenstate will be a 