In this paper, the local iterative Lie–Schwinger block-diagonalization method, introduced and developed in our previous work for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation, consisting of bounded interaction potentials of short range multiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms, which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, which serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step.
We developed a graph-based block-diagonalization (GBBD) method for the full configuration interaction Hamiltonian of molecular systems to efficiently calculate the exact eigenvalues of low-energy states. In this approach, the non-zero matrix elements of the Hamiltonian are represented as edges on a graph, which naturally decomposes into disconnected clusters. Each cluster corresponds to an independent block in the block-diagonalized form of the Hamiltonian. The eigenvalues in the low-energy sector were obtained by solving the eigenvalue problem for each block matrix and by solving a modified Hamiltonian subject to orthonormality constraints with respect to previously computed lower-energy eigenstates. We applied the GBBD method to linear hydrogen H chains ranging from H
2 to H
12. The results showed excellent agreement with exact ones, confirming both the accuracy and efficiency of the proposed method. Finally, we discussed several physical properties with respect to the number of H
2 molecules.
A formulation of numerical real-space renormalization groups for quantum many-body problems is presented and several algorithms utilizing this formulation are outlined. The methods are presented and demonstrated using S=1/2 and S=1 Heisenberg chains as test cases. The key idea of the formulation is that rather than keep the lowest-lying eigenstates of the Hamiltonian in forming a new effective Hamiltonian of a block of sites, one should keep the most significant eigenstates of the block density matrix, obtained from diagonalizing the Hamiltonian of a larger section of the lattice which includes the block. This approach is much more accurate than the standard approach; for example, energies for the S=1 Heisenberg chain can be obtained to an accuracy of at least ${10}^{\mathrm{\ensuremath{-}}9}$. The method can be applied to almost any one-dimensional quantum lattice system, and can provide a wide variety of static properties.
Symmetry plays a central role throughout physics, from Fourier analysis on discrete lattices to the classification of elementary particles in the standard model. In finite quantum systems, symmetries are often discrete, such as reflection and rotational symmetries. Here, we present the SIM-GRAPH method (Symmetric Ising Models - Graph Reduction And Projected Hamiltonians), which uses such symmetries to efficiently calculate ground-state observables of interacting quantum systems. By projecting all interactions onto a symmetric subset of the system, the method effectively reduces the number of sites, thus providing an exponential speedup over standard exact diagonalization. Furthermore, we introduce two extensions that broaden the applicability of the method and allow for a controlled trade-off between computational cost and accuracy.
The analysis of an open quantum system can be by far difficult if the dimension of the system Hilbert space is large or infinite. However, in some cases the dynamics on a finite-dimensional Hilbert space can be decomposed into a block-diagonal form, which simplifies the system structure. In this presentation, we will study several criteria for the complete reducibility and, in addition, a computational method for a basis of each simplified component to apply for the analysis of open quantum systems. An important point of these tools is that they are “effective” methods (one can complete the task in a finite number of steps).
Taming the exponential increase of the Hilbert space dimension with system size in the simulation of gapped quantum lattice models is of the highest relevance for understanding and designing exotic quantum materials, where nonlocal interactions are of particular interest. High-order linked-cluster expansions provide access the solution of the eigenvalue problem for the infinite system, yet rely on the computation of high-dimensional oscillatory lattice sums with a graph structure, only approachable with Monte Carlo methods so far. This work resolves this issue, rendering all required graph lattice sums, referred to as graph zeta functions for kernels involving power-laws, computable. The resulting method reduces the evaluation time for state-of-the art series expansions from tenthousands of core-hours to minutes. After factorizing the lattice sum over blocks, each block is evaluated by the cheapest available strategy depending on its treewidth
tw. Basic blocks admit analytic forms in terms of generalized zeta functions. Series-parallel blocks with
tw≤2 can be computed at linear cost in the number of graph nodes and in the size of the momentum grid using a semi-analytical algebra based on Epstein zeta functions and rapidly decaying Fourier series. Finally, for
tw>2, the method is combined with tensor-network bucket elimination yielding polynomial scaling of numerical work and memory in momentum grid size with exponents only growing with
tw rather than with the number of vertices. Through use of FFT, the full momentum grid is recovered at the cost of a single momentum evaluation. We provide a detailed analysis of the precision and runtime of our method against analytic and numerical benchmarks. We further reproduce published Monte Carlo data for the transverse-field Ising model on different 1D, 2D, and 3D lattices, obtaining full agreement.