Polynomial filter diagonalization of large Floquet unitary operators

Abstract

Periodically driven quantum many-body systems play a central role for our understanding of nonequilibrium phenomena. For studies of quantum chaos, thermalization, many-body localization and time crystals, the properties of eigenvectors and eigenvalues of the unitary evolution operator, and their scaling with physical system size LL are of interest. While for static systems, powerful methods for the partial diagonalization of the Hamiltonian were developed, the unitary eigenproblem remains daunting. % In this paper, we introduce a Krylov space diagonalization method to obtain exact eigenpairs of the unitary Floquet operator with eigenvalue closest to a target on the unit circle. Our method is based on a complex polynomial spectral transformation given by the geometric sum, leading to rapid convergence of the Arnoldi algorithm. We demonstrate that our method is much more efficient than the shift invert method in terms of both runtime and memory requirements, pushing the accessible system sizes to the realm of 20 qubits, with Hilbert space dimensions \geq 10^6≥106.

Avalanches and many-body resonances in many-body localized systems

“…We also note that the locality of this quantum circuit implies that the computational cost of applying it to a state is O(L2 L ). Thus while the Floquet unitary does not have a sparse matrix representation, it can be applied one gate at a time, and so it is compatible with algorithms that rely on matrix-free matrix-vector products like geometric sum filtering [76], which we use to access large system sizes. FIG.…”

“…For system sizes L ≤ 14, we use all eigenvalues of U obtained using exact diagonalization, and a number of disorder realizations which varies in the range 10 4 −4•10 4 . For L ≥ 16, we use the 50 eigenvalues closest to 1, calculated using geometric sum filtering [76] for 3000 − 6000 realizations. For the Hamiltonian model we average over the middle fifth of states in the spectrum and 8000 − 64, 000 disorder realizations for L ≤ 16 .…”

Stability of many-body localization in Kicked Ising model

“…This enables the Lanczos iteration to quickly converge to those eigenvectors. A polynomial which can be effectively used as the spectral filter for unitary operators was proposed in [108], and is simply a geometric sum:…”

“…This demonstrates the need of identifying quantum many-body systems that allow for a clearer demonstration of MBL than for the widely studied spin-1/2 XXZ chains . In this work we achieve this goal by performing large-scale numerical calculations for disordered Kicked Ising model (KIM) with state-of-the-art polynomially filtered exact diagonalization (POLFED) algorithm [62,108]. We identify ergodic, critical and MBL regimes by considering system size dependent disorder strengths W T X (L) and W * X (L) and quantitatively demonstrate that finite size effects at the ergodic to MBL crossover in KIM are significantly weaker than in the XXZ model.…”

“…Here, we consider higher values of W , up to a strong disorder limit in which j h j σ z j becomes a dominant term in U KIM . To find the eigenvectors |ψ n and the corresponding eigenvalues e iφn of U KIM , we use the POLFED algorithm [62] employing a geometric sum filtering [108] (see [115] for details). This allows us to obtain eigenstates |ψ n for system sizes L ≤ 20, significantly larger than for L ≤ 14 considered in earlier exact diagonalization studies of KIM [36,40].…”