Magnetic Resonance, Index Compression Maps and the Holstein-Primakoff Bosons: Polynomially Scaling Exact Diagonalization of Static Isotropic Multispin Hamiltonians
Matrix diagonalization has long been a setback in the numerical simulation of the magnetic resonance spectra of multispin systems since the dimension of the Hilbert space of such systems grows exponentially with the number of spins -- a problem commonly referred to as the of In this paper, we propose two mathematical instruments which, when harmoniously combined, could greatly help surmount to a fair degree and in a systematic manner the curse of dimensionality. These are: 1) the Holstein-Primakoff bosons and 2) what we have termed the index compression maps. These two allow a bijective mapping of (multi)spin states to integers. Their combination leads to the block diagonalization of the multispin Hamiltonian, thus a computationally exact way of diagonalizing the latter but which also scales polynomially with the number of spins. We also show that the eigenvectors and eigenvalues of the Liouvillian operator can be easily determined once those of the related multispin Hamiltonian are known. Interestingly, the method also enables an analytical characterization of the multispin Hilbert space -- a feat hardly attainable with other approaches.
We illustrate the method here by showing how a general static isotropic multispin Hamiltonian could be exactly diagonalized with very less computational cost. Nonetheless, we emphasize that the method could be applied to study numerous quantum systems defined on finite Hilbert spaces and embodied with at most pairwise interactions.
Matrix diagonalization has long been a setback in the numerical simulation of the magnetic resonance spectra of multispin systems since the dimension of the Hilbert space of such systems grows exponentially with the number of spins -- a problem commonly referred to as the of In this paper, we propose two mathematical instruments which, when harmoniously combined, could greatly help surmount to a fair degree and in a systematic manner the curse of dimensionality. These are: 1) the Holstein-Primakoff bosons and 2) what we have termed the index compression maps. These two allow a bijective mapping of (multi)spin states to integers. Their combination leads to the block diagonalization of the multispin Hamiltonian, thus a computationally exact way of diagonalizing the latter but which also scales polynomially with the number of spins. We also show that the eigenvectors and eigenvalues of the Liouvillian operator can be easily determined once those of the related multispin Hamiltonian are known. Interestingly, the method also enables an analytical characterization of the multispin Hilbert space -- a feat hardly attainable with other approaches.
We illustrate the method here by showing how a general static isotropic multispin Hamiltonian could be exactly diagonalized with very less computational cost. Nonetheless, we emphasize that the method could be applied to study numerous quantum systems defined on finite Hilbert spaces and embodied with at most pairwise interactions.
Matrix diagonalization has long been a setback in the numerical simulation of the magnetic resonance spectra of multispin systems since the dimension of the Hilbert space of such systems grows exponentially with the number of spins -- a problem commonly referred to as the "curse of dimensionality". In this paper, we propose two mathematical instruments which, when harmoniously combined, could greatly help surmount to a fair degree and in a systematic manner the curse of dimensionality. These are: 1) the Holstein-Primakoff bosons and 2) what we have termed the "index compression maps". These two allow a bijective mapping of (multi)spin states to integers. Their combination leads to the block diagonalization of the multispin Hamiltonian, thus a computationally exact way of diagonalizing the latter but which also reduces significantly the computational cost. We also show that the eigenvectors and eigenvalues of the Liouvillian operator can be easily determined once those of the related multispin Hamiltonian are known. Interestingly, the method also enables an analytical characterization of the multispin Hilbert space -- a feat hardly attainable with other approaches. We illustrate the method here by showing how a general static isotropic multispin Hamiltonian could be exactly diagonalized with very less computational cost. Nonetheless, we emphasize that the method could be applied to study numerous quantum systems defined on finite Hilbert spaces and embodied with at most pairwise interactions.
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.
The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert spaceHilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert space of the spin system with the number of spins: $$\dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right)$$ . Nowadays, complex Hermitian matricesHermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size si to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos methodFinite-Temperature Lanczos Method (FTLM), allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.
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.
Magnetic molecules promise progress in a wide variety of fields ranging from high density memory storage to quantum computing as well as magnetic cooling. In order to gain better insight this work aims at the study of theoretical models and evaluation of thermodynamic observables by implementing and testing a new numerical approximation method - the Finite-Temperature Lanczos-Method (FTLM). Magnetic centres of molecules are modelled as spins of fixed quantum number, position and interactions. The model is specified in a spin Hamiltonian consisting of three parts: Heisenberg exchange, single ion anisotropy and Zeeman interaction. Although finite, the dimension of the Hilbert space grows exponentially with the number of spin sites, so full diagonalisation of the Hamiltonian is rendered infeasible: Even for a small number of spins the Hilbert space dimension exceeds several millions, which is beyond exact diagonalisation techniques. In order to calculate observables yet, it is necessary to implement approximation methods. Following previous work, where the Hilbert space could be partitioned into invariant subspaces with respect to magnetic quantum numbers, now systems with anisotropy are studied. These anisotropic terms generally do not commute with total spin operators, therefore partitioning no longer helps reducing the problem size. There are several techniques for approximate solutions to eigenvalue equations, among these adaptations of the original Lanczos method. As a Krylov subspace method, its main advantage is that the most complex operations required are matrix-vector multiplications. When matrix elements are calculated on-the-fly, the most economical version requires storage of only two vectors. The second ingredient for FTLM is a typicality based Monte Carlo approach: Traces are estimated as an average over a tiny set of initial vectors compared to the full Hilbert space. First tests revealed a major flaw due to numerical instability of traces that normally are zero. By partitioning with respect to magnetic quantum numbers, especially exploiting their inversion symmetry, this problem was unknowingly dodged in previous calculations. It was solved while preserving the advantage of minimal computational effort: An arbitrary set of initial vectors would regain this time-reversal symmetry only, if a substantial number of initial vectors where used. Instead, a doubling of the initial vectors by inverting them with respect to magnetic quantum numbers yields the desired effect. Time-consuming evaluations of matrix elements were also avoided by calculating the magnetisation as a difference quotient. Finally FTLM is implemented and tested for several magnetic molecules. Analogues of the hour glass molecules synthesised by Glaser et al. were first model systems, showing good agreement of exact diagonalisation results and approximations. The method can be applied to any Hamiltonian, given that matrix-vector products in the underlying Hilbert space can be calculated. Here limiting factors are memory usage and computing time. As a proof of principle observables in a Hilbert space consisting of d=100,000,000 states were calculated using the parameter set suggested by Mazurenko et al. for the Mn12-acetate molecule. Single crystal as well as powder averaged results are compared to experimental data. So this previously impossible calculation is now executed within hours by use of FTLM.
The procedure for simulating the nuclear magnetic resonance spectrum linked to the spin system of a molecule for a certain nucleus entails diagonalizing the associated Hamiltonian matrix. As the dimensions of said matrix grow exponentially with respect to the spin system's atom count, the calculation of the eigenvalues and eigenvectors marks the performance of the overall process. The aim of this paper is to provide a formalization of the complete procedure of the simulation of a spin system's NMR spectrum while also explaining how to diagonalize the Hamiltonian matrix with a quantum computer, thus enhancing the overall process's performance. Two well-known quantum algorithms for calculating the eigenvalues of a matrix are analyzed and put to the test in this context: quantum phase estimation and the variational quantum eigensolver. Additionally, we present simulated results for the later approach while also addressing the hypothetical noise found in a physical quantum computer.