Eigenstate Thermalization in Dual Unitary Quantum Circuits

热力学平衡态对孤立量子系统的成功描述引发了这样一个问题：在幺正时间演化下，这些系统如何从典型的纯初态热化。本征态热化假设（ETH）用描述宏观观测量矩阵元结构的数学分析给出了答案。尽管数值研究为ETH在一般多体量子系统中的存在提供了强有力的证据，但很少有分析证明给定系统确实满足ETH。在这个项目中，我建议研究双幺正量子电路中的本征态热化，并严格证实ETH及其含义。这样的量子电路为一般的多体量子系统提供了最小模型，并允许对其突出特征进行分析描述。特别是双酉电路模型，表现出额外的时空对称性，适合于精确计算，因此是ETH分析确认的绝佳候选者。为此，我计划（i）数值研究双幺正量子电路中的本征态热化，（ii）建立它们的量子动力学的解析描述，（iii）确认局部可观测量的ETH近似，（iv）研究本征态的纠缠特性，以及（v）将双幺正量子电路的特性扩展到更一般的多体系统。

The successful description of isolated quantum systems by thermodynamic equilibrium states has triggered the question how those systems thermalize from a typical pure initial state under unitary time evolution. The eigenstate thermalization hypothesis (ETH) conjectures an answer in terms of a mathematical ansatz which describes the structure of matrix elements of macroscopic observables. Although numerical studies provide strong evidence for ETH in generic many-body quantum systems, analytical proofs that a given system indeed fulfills ETH are rare. In this project I propose to study eigenstate thermalization in dual-unitary quantum circuits and to rigorously confirm ETH and its implications. Such quantum circuits provide minimal models for generic many-body quantum systems and allow for an analytical description of their prominent features. In particular dual-unitary circuit models, which exhibit an additional space-time symmetry, are suitable for exact calculations and thus are excellent candidates for an analytic confirmation of ETH. To this end I plan to (i) numerically investigate eigenstate thermalization in dual-unitary quantum circuits, (ii) establish an analytical description of their quantum dynamics, (iii) confirm the ETH ansatz for local observables, (iv) study entanglement properties of eigenstates, and (v) extend the properties of dual-unitary quantum circuits to more generic many-body systems.