Iterative methods that operate with the full Hamiltonian matrix in the untrimmed Hilbert space of a finite system continue to be important tools for the study of one- and two-dimensional quantum spin models, in particular in the presence of frustration. To reach sensible system sizes such numerical calculations heavily depend on the use of symmetries. We describe a divide-and-conquer strategy for implementing translation symmetries of finite spin clusters, which efficiently uses and extends the "sublattice coding" of H. Q. Lin [Phys. Rev. B 42, 6561 (1990)]. With our method, the Hamiltonian matrix can be generated on-the-fly in each matrix vector multiplication, and problem dimensions beyond 10^{11} become accessible.
Iterative methods that operate with the full Hamiltonian matrix in the untrimmed Hilbert space of a finite system continue to be important tools for the study of one- and two-dimensional quantum spin models, in particular in the presence of frustration. To reach sensible system sizes such numerical calculations heavily depend on the use of symmetries. We describe a divide-and-conquer strategy for implementing translation symmetries of finite spin clusters, which efficiently uses and extends the "sublattice coding" of H. Q. Lin [Phys. Rev. B 42, 6561 (1990)]. With our method, the Hamiltonian matrix can be generated on-the-fly in each matrix vector multiplication, and problem dimensions beyond 10^{11} become accessible.
Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry CN in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2) can be used, where the Hamiltonian matrix is block structured according to the total spin and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both CN and SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry-reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.
The complexity of quantum many-body problems scales exponentially with the size of the system, rendering any finite-size scaling analysis a formidable challenge. This is particularly true for methods based on the full representation of the wave function, where one simply accepts the enormous Hilbert space dimensions and performs linear algebra operations, e.g., for finding the ground state of the Hamiltonian. If the system satisfies an underlying symmetry where an operator with a degenerate spectrum commutes with the Hamiltonian, it can be block-diagonalized, thus reducing the complexity at the expense of additional bookkeeping. At the most basic level required for Krylov space techniques (like the Lanczos algorithm), it is necessary to implement a matrix-vector product of a block of the Hamiltonian with arbitrary block-wavefunctions, potentially without holding the Hamiltonian block in memory. An efficient implementation of this operation requires the calculation of the position of an arbitrary basis vector in the canonical ordering of the basis of the block. We present here an elegant and powerful, multi-dimensional approach to this problem for the U(1) U ( 1 ) symmetry appearing in problems with particle number conservation. Our divide-and-conquer algorithm uses multiple subsystems and hence generalizes previous approaches to make them scalable. In addition to the theoretical presentation of our algorithm, we provide DanceQ, a flexible and modern – header only – C++20 implementation to manipulate, enumerate, and map to its index any basis state in a given particle number sector as open source software under https://DanceQ.gitlab.io/danceq.
The exploration of neural network quantum states has become widespread in studies of complicated quantum many-body systems. However, achieving high precision remains challenging due to the exponential growth of Hilbert space size and the intricate sign structures. Utilizing symmetries of the physical system, we propose a method to evaluate and sample the variational Ansatz within a symmetric subspace. This approach isolates different symmetry sectors, reducing the relevant Hilbert space size by a factor approximately proportional to the size of the symmetry group. It is inspired by exact diagonalization techniques and the work of Choo et al., [Phys. Rev. Lett. 121, 167204 (2018)]. We validate our method using the frustrated spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg chain, and we compare its performance to the case without symmetrization. The results indicate that our symmetric subspace approach achieves a substantial improvement over the full Hilbert space on optimizing the Ansatz, reducing the energy error by orders of magnitude. We also compare the results on degenerate eigenstates with different quantum numbers, highlighting the advantage of operating within a smaller Hilbert subspace.
The complexity of quantum many-body problems scales exponentially with the size of the system, rendering any finite-size scaling analysis a formidable challenge. This is particularly true for methods based on the full representation of the wave function, where one simply accepts the enormous Hilbert space dimensions and performs linear algebra operations, e.g., for finding the ground state of the Hamiltonian. If the system satisfies an underlying symmetry where an operator with a degenerate spectrum commutes with the Hamiltonian, it can be block-diagonalized, thus reducing the complexity at the expense of additional bookkeeping. At the most basic level required for Krylov space techniques (like the Lanczos algorithm), it is necessary to implement a matrix-vector product of a block of the Hamiltonian with arbitrary block-wavefunctions, potentially without holding the Hamiltonian block in memory. An efficient implementation of this operation requires the calculation of the position of an arbitrary basis vector in the canonical ordering of the basis of the block. We present here an elegant and powerful, multi-dimensional approach to this problem for the U(1) U ( 1 ) symmetry appearing in problems with particle number conservation. Our divide-and-conquer algorithm uses multiple subsystems and hence generalizes previous approaches to make them scalable. In addition to the theoretical presentation of our algorithm, we provide DanceQ, a flexible and modern – header only – C++20 implementation to manipulate, enumerate, and map to its index any basis state in a given particle number sector as open source software under https://DanceQ.gitlab.io/danceq.
In spin lattices, frustration happens when there exists a competition between different terms of the Hamiltonian, which cannot be simultaneously minimized in all local physical bases. Frustration can destabilize the bipartite Neel order, which is the conventional groundstate of antiferromagnetic Heisenberg-type models in two dimensions. Hence, the focus of two-dimensional quantum magnetism studies has been directed toward elucidating the zero-temperature properties of Heisenberg-type models on the Archimedean lattices that exhibit geometrical frustration. This type of frustration is of great interest since it can lead to exotic forms of the quantum matter, such as topological spin-liquid (SL) and many-sublattice long-range-ordered phases. The spin-1/2 triangular Heisenberg model (THM) is a prototypical model with geometrical frustration, which has the highest coordination number of all Archimedean lattices. However, there exist two major, distinct difficulties for numerical methods to efficiently determine the groundstate properties of the THM: the immense dimension of the Hilbert space and the geometrical frustration phenomenon. In this thesis, to deal with such difficulties, we employ the state-of-the- art non-Abelian matrix product states (MPS) and density matrix renormalization group (DMRG) methods. We exploit the symmetries of the Hamiltonian to reduce the working dimension of the Hilbert space and, to lessen the effects of the geometrical frustration, we depend on the local entanglement (area-law) nature of the MPS as it scales with the size of a lattice subspace with reduced spatial dimensions. In such a way, we thoroughly examine the phase diagram of the THM with nearest and next-nearest neighbor (NN and NNN) exchange couplings on finite- and infinite-length cylinders of widths up to 12 sites. In addition, we develop new numerical tools to analyze topological and symmetry-broken phases in the context of the SU(2)-symmetric translation-invariant MPS algorithm. We establish that on infinite cylinders, the anyonic sectors of a topological order can be classified through the fractionalization of global symmetries and degeneracy patterns of the entanglement spectrum (ES). On the other hand, we detect symmetry-breaking phase transitions by a combination of the correlation lengths and second and fourth cumulants of the magnetic order parameters, even though symmetry implies that the order parameter itself is strictly zero. Furthermore, the appearance of Nambu-Goldstone modes in the excitation spectrum can be detected using “tower of states” (TOS) levels in the momentum-resolved ES. By applying the new tools to the THM, we discover a variety of exotic groundstates such as a Z2-gauge toric-code-type topological order (with vison and spinon excitations), a NNN Majumdar-Gosh, and a long-range 120◦-ordered state.
We present an algorithm that computes the action of a generic spin Hamiltonian on a state vector on the fly, entirely avoiding explicit matrix assembly. This is achieved through mixed-radix indexing of the full tensor-product basis, which translates local spin operations into simple integer offsets. The result is an explicit framework for evaluating single- and two-site terms across arbitrary spin lattices, including mixed-spin systems. Our construction bridges the basis-indexing logic familiar from exact diagonalization with the matrix-free state-update philosophy of address-based frameworks. By writing the indexing logic in closed form, a single uniform loop applies to every site regardless of its local Hilbert-space dimension. The method is parallelizable and memory-conserving, and can be extended to restricted basis or truncated bosonic levels.