论文标题
用于研究混乱本征态物理可观察物的多项式时间算法
A polynomial time algorithm for studying physical observables in chaotic eigenstates
论文作者
论文摘要
We introduce an algorithm, the Orthogonal Operator Polynomial Expansion (OOPEX), to approximately compute expectation values in energy eigenstates at finite energy density of non-integrable quantum many-body systems with polynomial effort, whereas exact diagonalization (ED) of the Hamiltonian $H$ is exponentially hard. OOPEX依赖于本征态热假设，该假设猜测，在此类系统中，物理可观察物的本征状期望值与本征征能（以及其他宏观保守量（如果有）（如果有的话）），并通过重复乘以乘以$ h $ $ h $ y的能量来计算它们。该假设保证了本系列的前几个术语只能显着贡献。我们进一步表明，从某种意义上说，OOPEX是基于$ h $的系列扩展的最佳算法，因为它避免计算困扰其他类似算法的状态的多体密度。然后，我们毫不恰当地认为，在运营商的Fock空间中工作，而不是按通常的状态工作，可以通过计算资源来产生收敛的结果，这些计算资源以$ n $进行多项式扩展。我们通过将oopex应用于不可融合的iSing链，并与ED和高温扩展（HTX）结果进行比较来证明多项式缩放。 OOPEX提供了比ED和HTX DO更大的$ N $的访问权限，这有助于克服有限尺寸的效果，这些效果困扰着其他方法以提取混乱的特征态中的相关长度。此外，对大型系统的访问允许测试最近的猜想，即混沌特征态的Renyi熵（如果Renyi索引$> 1 $）具有正弯曲，并且我们会发现令人鼓舞的证据。
We introduce an algorithm, the Orthogonal Operator Polynomial Expansion (OOPEX), to approximately compute expectation values in energy eigenstates at finite energy density of non-integrable quantum many-body systems with polynomial effort, whereas exact diagonalization (ED) of the Hamiltonian $H$ is exponentially hard. The OOPEX relies on the eigenstate thermalization hypothesis, which conjectures that eigenstate expectation values of physical observables in such systems vary smoothly with the eigenstate energy (and other macroscopic conserved quantities, if any), and computes them through a series generated by repeated multiplications, rather than diagonalization, of $H$ and whose successive terms oscillate faster with the energy. The hypothesis guarantees that only the first few terms of this series contribute appreciably. We further show that the OOPEX, in a sense, is the most optimum algorithm based on series expansions of $H$ as it avoids computing the many-body density of states which plagues other similar algorithms. Then, we argue non-rigorously that working in the Fock space of operators, rather than that of states as is usually done, yields convergent results with computational resources that scale polynomially with $N$. We demonstrate the polynomial scaling by applying the OOPEX to the non-integrable Ising chain and comparing with ED and high-temperature expansion (HTX) results. The OOPEX provides access to much larger $N$ than ED and HTX do, which facilitates overcoming finite-size effects that plague the other methods to extract correlation lengths in chaotic eigenstates. In addition, access to large systems allows testing a recent conjecture that the Renyi entropy of chaotic eigenstates has positive curvature if the Renyi index $>1$, and we find encouraging supporting evidence.
