多体量子系统中的亚稳态理论
Theory of metastable states in many-body quantum systems
AI总结：
该研究提出多体量子系统亚稳态的数学理论，证明短程纠缠亚稳态是微扰相近哈密顿量的本征态，推导假真空寿命下界，在多个模型中找到亚稳态，建立多类量子问题的关联。
摘要：
我们提出了一套关于有限维希尔伯特空间封闭多体量子系统中亚稳纯态的数学理论。给定一个哈密顿量，当所有足够局域的算符要么使该纯态保持稳定，要么提升其平均能量时，该纯态被定义为亚稳态。我们证明了短程纠缠的亚稳态必然是一个微扰相近的哈密顿量的本征态（疤痕态）。对于任意哈密顿量的亚稳本征态，在存在微扰的情况下，我们证明了预热化行为的存在：局域关联函数的衰减速率受一个时间尺度的约束，该时间尺度在亚稳半径的倒数中具有非微扰的长度，而非遵循费米黄金定则。受这一通用理论的启发，我们证明了特定d维量子模型中假真空的寿命增长速度至少与exp(ε^{-d})一样快，其中ε→0是假真空的相对能量密度；这一下界首次与利用量子场论得到的显式计算结果相匹配。我们在PXP模型中识别出有限能量密度下的亚稳态，同时在“螺旋”自旋链和二维Ising模型中发现了指数级数量的亚稳态。我们这种本质上是量子的形式体系揭示了包括预热化、鲁棒量子疤痕态和量子成核理论在内的诸多问题之间的精确关联，且适用于不存在已知半经典和/或场论极限的系统。
英文摘要：
We present a mathematical theory of metastable pure states in closed many-body quantum systems with finite-dimensional Hilbert space. Given a Hamiltonian, a pure state is defined to be metastable when all sufficiently local operators either stabilize the state, or raise its average energy. We prove that short-range entangled metastable states are necessarily eigenstates (scars) of a perturbatively close Hamiltonian. Given any metastable eigenstate of a Hamiltonian, in the presence of perturbations, we prove the presence of prethermal behavior: local correlation functions decay at a rate bounded by a time scale nonperturbatively long in the inverse metastability radius, rather than Fermi's Golden Rule. Inspired by this general theory, we prove that the lifetime of the false vacuum in certain $d$-dimensional quantum models grows at least as fast as $\exp(ε^{-d} )$, where $ε\rightarrow 0$ is the relative energy density of the false vacuum; this lower bound matches, for the first time, explicit calculations using quantum field theory. We identify metastable states at finite energy density in the PXP model, along with exponentially many metastable states in "helical" spin chains and the two-dimensional Ising model. Our inherently quantum formalism reveals precise connections between many problems, including prethermalization, robust quantum scars, and quantum nucleation theory, and applies to systems without known semiclassical and/or field theoretic limits.
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