A polynomial time algorithm for studying physical observables in chaotic eigenstates

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.