子系统能量统计中的面积律和普适性
Area law and universality in the statistics of subsystem energy
摘要：
我们引入子系统能量的Rényi熵作为一个自然量，它可以很好地模拟纠缠熵的行为，并且可以定义为所有的量子多体系统。为此，考虑一个处于基态的量子链，然后取该系统的一个子域，使其具有自然截断的哈密顿量。由于总哈密顿量与截断哈密顿量不对易，子系统可以以不同的概率处于其本征能之一。利用全局能量本征态在局部能量本征基中局部接近对角的事实，我们论证了对于带隙系统，这些概率的Rényi熵遵循面积律。当系统处于临界点时，Rényi熵遵循具有普适系数的对数行为。因此，我们的量不仅检测相变，但也确定了普遍性类的临界点。此外，我们表明，最大定义的概率是非常接近的最大施密特系数。我们通过定义{ \it{truncated} }香农熵来量化这一点，其值几乎与{ \it{truncated} }冯诺依曼纠缠熵无法区分。与纠缠熵相比，我们的量具有与自然可观测量（子系统哈密顿量）相关联的优点，该可观测量可以容易地定义（并且可能测量）所有短程相互作用系统。我们支持我们的论点进行详细的数值计算的横向场XY链。
英文摘要：
We introduce Rényi entropy of a subsystem energy as a natural quantity which closely mimics the behavior of the entanglement entropy and can be defined for all the quantum many body systems. For this purpose, consider a quantum chain in its ground state and then, take a subdomain of this system with natural truncated Hamiltonian. Since the total Hamiltonian does not commute with the truncated Hamiltonian, the subsystem can be in one of its eigenenergies with different probabilities. Using the fact that the global energy eigenstates are locally close to diagonal in the local energy eigenbasis, we argue that the Rényi entropy of these probabilities follows an area law for the gapped systems. When the system is at the critical point, the Rényi entropy follows a logarithmic behavior with a universal coefficient. Consequently, our quantity not only detects the phase transition but also determines the universality class of the critical point. Moreover we show that the largest defined probabilities are very close to the biggest Schmidt coefficients. We quantify this by defining a {\it{truncated}} Shannon entropy which its value is almost indistinguishable from the {\it{truncated}} von Neumann entanglement entropy. Compare to the entanglement entropy, our quantity has the advantage of being associated to a natural observable (subsystem Hamiltonian) that can be defined (and probably measured) easily for all the short-range interacting systems. We support our arguments by detailed numerical calculations performed on the transverse field XY-chain.
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