Disorder and interactions can lead to the breakdown of statistical mechanics in certain quantum systems, a phenomenon known as many-body localization (MBL). Much of the phenomenology of MBL emerges from the existence of $\ell \mathrm{bits}$, a set of conserved quantities that are quasilocal and binary (i.e., possess only $\pm 1$ eigenvalues). While MBL and $\ell \mathrm{bits}$ are known to exist in one-dimensional systems, their existence in dimensions greater than one is a key open question. To tackle this question, we develop an algorithm that can find approximate binary $\ell \mathrm{bits}$ in arbitrary dimensions by adaptively generating a basis of operators in which to represent the $\ell \mathrm{bit}$. We use the algorithm to study four models: the one-, two-, and three-dimensional disordered Heisenberg models and the two-dimensional disordered hard-core Bose-Hubbard model. For all four of the models studied, our algorithm finds high-quality $\ell \mathrm{bits}$ at large disorder strength and rapid qualitative changes in the distributions of $\ell \mathrm{bits}$ in particular ranges of disorder strengths, suggesting the existence of MBL transitions. These transitions in the one-dimensional Heisenberg model and two-dimensional Bose-Hubbard model coincide well with past estimates of the critical disorder strengths in these models, which further validates the evidence of MBL phenomenology in the other two- and three-dimensional models we examine. In addition to finding MBL behavior in higher dimensions, our algorithm can be used to probe MBL in various geometries and dimensionality.

中文翻译：

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二维和三维多体定位的数值证据

无序和相互作用会导致某些量子系统中的统计力学崩溃，这种现象称为多体定位（MBL）。MBL的许多现象学源于$\ell 位$，一组准局部和二进制守恒量（即，仅具有 $\pm \mathrm{1个}$特征值）。而MBL和$\ell 位$已知存在于一维系统中，它们的存在大于一维是一个关键的未解决的问题。为了解决这个问题，我们开发了一种算法，可以找到近似的二进制$\ell 位$ 通过适应性地生成一个表示算子的运算符的基础，在任意维度上 $\ell \mathrm{少量}$。我们使用该算法研究四个模型：一维，二维和三维无序Heisenberg模型以及二维无序硬核Bose-Hubbard模型。对于所研究的所有四个模型，我们的算法都能找到高质量的$\ell 位$ 在较大的无序强度和快速的质变分布中 $\ell 位$特别是无序强度的范围，表明存在MBL转换。一维海森堡模型和二维Bose-Hubbard模型中的这些转变与这些模型中关键疾病强度的以往估计值非常吻合，这进一步验证了我们在其他二维模型和三维模型中MBL现象学的证据。检查。除了发现更高尺寸的MBL行为外，我们的算法还可用于探测各种几何形状和尺寸的MBL。