Investigating non-integrable quantum many-body systems with sub-exponential computational effort

Non-integrable quantum systems - systems with no or few conserved quantities - are generically exponentially hard to solve. Recent developments have proven, at least for small system sizes, that these systems satisfy the eigenstate thermalization hypothesis (ETH). Specifically, their physical properties such as expectation values of simple observables and few-point correlation functions, depend smoothly on macroscopic variables such as the energy density, particle density etc. This behavior explains, qualitatively, how statistical mechanics and ergodicity emerges from quantum mechanical laws for such systems. In this talk, an algorithm will be presented that enables quantitative predictions based on the ETH. In particular, it can simulate finite temperature properties of non-integrable systems with sub-exponential effort by exploiting the ETH. By expressing an eigenstate density matrix as truncatable series of orthogonal polynomials of the Hamiltonian, it will require only sparse matrix multiplication instead of matrix diagonalization. Moreover, it will store sparse matrices as vectors in the Fock space of operators, which reduces storage requirements from exponential to polynomial. Benchmarks between the algorithm and exact diagonalization on prototypical models will be shown.