论文标题
Majoraana疤痕作为团体单人
Majorana Scars as Group Singlets
论文作者
论文摘要
在某些量子多体系统中，希尔伯特空间分解为一个大型的千古扇区，并且疤痕子空间要小得多。有人建议[Arxiv：2007.00845]，这两个部门可以通过其在系统大小随系统大小增长的大型群体下的转化特性来区分（这不是汉密尔顿的对称性）。量子多体疤痕在该组下是不变的，而所有其他状态却不是。在这里，我们将此想法应用于每个站点中包含$ M $ Majorana Fermions的晶格系统。 $ n $站点的希尔伯特空间可能是根据o $ $（n）\ times $ o $ $（m）$组分解的，而伤疤是So $（n）$ singlets。对于任何$ m $，都有两个疤痕家族。我们称之为$η$状态的其中之一是o $ $（n）$的对称性。另一个$ζ$状态具有SO $（N）$不变性。对于$ m = 4 $，我们的施工在与本地互动的晶格上旋转至$ 1/2 $ fermions，以前的家庭是$ n+1 $ $ $ $ $ $ $ $ $ hum-pairing的状态，而后者则是最大旋转的$ n+1 $。我们将这种结构推广到$ m> 4 $。对于$ m = 6 $，我们为疤痕状态展示明确的公式，并使用它们分析地计算两部分纠缠熵。对于大$ n $，它随着子系统的大小而对数增长。我们提出了一个普遍的论点，即任何群体不变的疤痕都应具有比典型状态小的参数熵。我们发现的疤痕的能量一般不是等距的，但可以通过选择哈密顿参数来制作。对于$ m> 6 $，我们发现与当地的汉密尔顿人一起，疤痕通常具有某些变性。可以通过添加非局部相互作用项来使疤痕谱成为千古谱。我们得出了每个疤痕家族的尺寸，并表明疤痕可能会对小$ n $的州密度有很大的贡献。
In some quantum many-body systems, the Hilbert space breaks up into a large ergodic sector and a much smaller scar subspace. It has been suggested [arXiv:2007.00845] that the two sectors may be distinguished by their transformation properties under a large group whose rank grows with the system size (it is not a symmetry of the Hamiltonian). The quantum many-body scars are invariant under this group, while all other states are not. Here we apply this idea to lattice systems containing $M$ Majorana fermions per site. The Hilbert space for $N$ sites may be decomposed under the action of the O$(N)\times$O$(M)$ group, and the scars are the SO$(N)$ singlets. For any even $M$ there are two families of scars. One of them, which we call the $η$ states, is symmetric under the group O$(N)$. The other, the $ζ$ states, has the SO$(N)$ invariance. For $M=4$, where our construction reduces to spin-$1/2$ fermions on a lattice with local interactions, the former family are the $N+1$ $η$-pairing states, while the latter are the $N+1$ states of maximum spin. We generalize this construction to $M>4$. For $M=6$ we exhibit explicit formulae for the scar states and use them to calculate the bipartite entanglement entropy analytically. For large $N$, it grows logarithmically with the sub-system size. We present a general argument that any group-invariant scars should have the entanglement entropy that is parametrically smaller than that of typical states. The energies of the scars we find are not equidistant in general but can be made so by choosing Hamiltonian parameters. For $M>6$ we find that with local Hamiltonians the scars typically have certain degeneracies. The scar spectrum can be made ergodic by adding a non-local interaction term. We derive the dimension of each scar family and show the scars could have a large contribution to the density of states for small $N$.
