arXiv 2009.138732020-09quant-phcond-mat.othercond-mat.quant-gas

含时与不含时量子多体哈密顿算符之间的映射

A map between time-dependent and time-independent quantum many-body Hamiltonians

Oleksandr Gamayun, Oleg Lychkovskiy

备注：: to be published in Proceedings of the Steklov Institute of Mathematics
发表信息：: Proceedings of the Steklov Institute of Mathematics, 313, 41 (2021)
DOI：: 10.1134/S008154382102005X

摘要：

给定一个与时间无关的哈密顿量$\widetilde H$，人们可以通过规范变换$H_t=U_t \widetilde H \, U^\dagger_t-i\, U_t\, \partial_t U_t^\dagger$构造一个与时间相关的哈密顿量$H_t$。这里$U_t$是与相应薛定谔方程的解相关的么正变换。在多体情况下，人们通常对具有少体（通常最多两体）相互作用的哈密顿量感兴趣。我们把这种哈密顿量称为“物理的”。我们制定$U_t$的充分条件，确保$H_t$是物理的，只要$\widetilde H$是物理的（反之亦然）。这样我们就得到了一种寻找这样的物理哈密顿量对$H_t$，$\widetilde H$的一般方法，即由$H_t$支配的受驱动多体动力学可以归结为由于与时间无关的$\widetilde H$而引起的猝灭动力学。我们将这种方法应用到一些多体系统。首先回顾了具有各向同性海森堡相互作用和任意含时磁场的自旋系统到无磁场的非含时系统的映射[F.延湖，澳-地扬湾，澳-地Li，Phys. Lett. A 251，289（1999）; Phys. Lett. A 259，207（1999）]。然后，我们证明，本质上相同的规范变换消除了任意的时间相关的磁场从相互作用的费米子系统。此外，我们应用该方法的量子伊辛自旋系统和自旋耦合到玻色子环境。我们还讨论了一个更一般的情况下，$\widetilde H = \widetilde H_t$是时间相关的，但动态可积。

英文摘要：

Given a time-independent Hamiltonian $\widetilde H$, one can construct a time-dependent Hamiltonian $H_t$ by means of the gauge transformation $H_t=U_t \widetilde H \, U^\dagger_t-i\, U_t\, \partial_t U_t^\dagger$. Here $U_t$ is the unitary transformation that relates the solutions of the corresponding Schrodinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as "physical". We formulate sufficient conditions on $U_t$ ensuring that $H_t$ is physical as long as $\widetilde H$ is physical (and vice versa). This way we obtain a general method for finding such pairs of physical Hamiltonians $H_t$, $\widetilde H$ that the driven many-body dynamics governed by $H_t$ can be reduced to the quench dynamics due to the time-independent $\widetilde H$. We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to the time-independent system without a magnetic field [F. Yan, L. Yang, B. Li, Phys. Lett. A 251, 289 (1999); Phys. Lett. A 259, 207 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where $\widetilde H = \widetilde H_t$ is time-dependent but dynamically integrable.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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