Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.17429

GPU-accelerated finite-temperature Lanczos method for Heisenberg spin systems with non-Abelian permutation symmetries

Shadan Ghassemi Tabrizi, Thomas D. Kühne

Published Jul 19, 2026• 0 citations
arXivHTMLPDF

Abstract

We extend a recent GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians, which uses only total-magnetization symmetry, to the full permutation symmetry, including non-Abelian point and space groups. For highly symmetric clusters such as icosahedral polyhedra, the multidimensional irreducible representations (irreps) of the permutation group reduce the block dimensions substantially further than any Abelian subgroup. Here, we provide a matrix-free formalism (avoiding explicit storage of the projected Hamiltonian) for using such symmetries. The implementation applies equally to Abelian groups, retaining the advantages of GPU acceleration. Several cluster topologies are built in; beyond these, symmetry groups can be supplied as site-permutation generators, with irrep matrices computed automatically, so that user-defined clusters run without any code changes. We demonstrate the approach in production runs on a single NVIDIA B200 accelerator for the s=3/2 Heisenberg antiferromagnet on the dodecahedron (Hilbert-space dimension 4^20 ~ 1.1x10^12) and for the s=1/2 square lattice of 5x8 sites (M=0 dimension 1.4x10^11). With R=24 random vectors per symmetry block and 60 Lanczos steps, the dodecahedron campaign (with the largest iterated symmetry block having a dimension of 3.6x10^9) requires about 50 GPU-hours. The code is openly available under the Apache-2.0 license at https://github.com/ghasdeke/ftlm-pg-gpu, archived at DOI: 10.5281/zenodo.21445872.

DOI
10.48550/arxiv.2607.17429

Related papers

GPU-accelerated finite-temperature Lanczos method for spin Hamiltonians

Shadan Ghassemi Tabrizi, Thomas D. Kühne

We present a GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians that targets workstation hardware rather than distributed-memory clusters. The Hamiltonian action is evaluated matrix-free in a row-wise gather formulation. We introduce and compare two state-to-index strategies: a compressed lookup table (CLT), which reduces lookup memory by a factor of 16 relative to a full table while retaining a fixed, branch-light access pattern, and a GPU-adapted combinatorial-ranking scheme that removes the lookup table altogether. Numerical tests against FP64 CPU references show that FP32 GPU arithmetic changes heat capacities and magnetic susceptibilities by amounts several orders of magnitude below the stochastic uncertainty of the FTLM trace estimator at typical sample sizes. Benchmarks show speedups of up to about one order of magnitude over optimized multicore CPU calculations and enable Hilbert-space sectors of dimension ~10^8 on a single workstation GPU. The MATLAB/CUDA implementation, including example input files and benchmark scripts, is openly available at https://github.com/ghasdeke/ftlm-gpu (archived at DOI: 10.5281/zenodo.20378647) under the Apache-2.0 license.

arXiv:2605.26261•May 25, 2026•0 citations

Implementation of FTLM for full Hilbert space investigation of anisotropic quantum spin systems

Oliver Hanebaum

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.

•Jan 1, 2018•0 citations

Symmetry adapted finite-cluster solver for quantum Heisenberg model in two dimensions: a real-space renormalization approach

Vl. E. Sinitsyn, Irina Gennad'evna Bostrem, Alexander S. Ovchinnikov

We present a quantum cluster solver for the spin- S Heisenberg model on a two-dimensional lattice. The formalism is based on the real-space renormalization procedure and uses the lattice point group-theoretical analysis and non-Abelian SU (2) spin symmetry technique. The exact diagonalization procedure is used twice at each renormalization group step. The method is applied to the spin-half antiferromagnet on a square lattice, and a calculation of local observables is demonstrated. A symmetry-based truncation procedure is suggested and verified numerically.

arXiv:cond-mat/0612350•Journal of Physics A Mathematical and Theoretical•Jan 9, 2007•14 citations

The Study of Model Hamiltonians on Finite Clusters.

Duan Chen

We present two methods to study model Hamiltonians defined on clusters. One is the symmetrization of the Hilbert space basis according to irreducible representations of the symmetry group of the Hamiltonian. The other one adapts the Lanczos diagonalization method directly to the second quantized form of the Hamiltonian studied. By the symmetrization of the basis, one takes advantage of the symmetries of the Hamiltonian, and uses the group theory to find the invariant subspaces of the Hamiltonian in the complete Hilbert space. Then one needs only to work within the invariant subspaces. This method cuts down substantially the size of the matrix representation of the Hamiltonian to be diagonalized. Therefore, one could study relatively larger systems. As a result, all the eigenvectors and eigenvalues are obtained. Consequently, one could calculate thermodynamical quantities and various correlation functions, such as, the spin correlation, the charge correlation, the superconductivity pairing correlation, and the photoemission spectra. One the other hand, the Lanczos method, mentioned here in particular, is a specially developed computer program for the study of the model Hamiltonians. This program uses the key feature of the Lanczos method in diagonalization, allows one to use the second quantized form of the Hamiltonian directly without pregeneration or storing the matrix elements of the Hamiltonian. Two methods could be combined to take the full advantage of the symmetries of the Hamiltonian by input of the symmetrized wavefunction as the starting trial wavefunction in the Lanczos method. Here we illustrate the above two methods by applying to the Hubbard, and extended Hubbard models on a simple cubic cluster, and the lattice Anderson model on the two dimensional square lattice. Other interesting results for different systems are also briefly discussed here.

Civil War Book Review•Jan 1, 1989•0 citations

Exact Diagonalization of Heisenberg SU ( N ) Models

Pierre Nataf, Frédéric Mila

Building on advanced results on permutations, we show that it is possible to construct, for each irreducible representation of SU(N), an orthonormal basis labeled by the set of standard Young tableaux in which the matrix of the Heisenberg SU(N) model (the quantum permutation of N-color objects) takes an explicit and extremely simple form. Since the relative dimension of the full Hilbert space to that of the singlet space on n sites increases very fast with N, this formulation allows us to extend exact diagonalizations of finite clusters to much larger values of N than accessible so far. Using this method, we show that, on the square lattice, there is long-range color order for SU(5), spontaneous dimerization for SU(8), and evidence in favor of a quantum liquid for SU(10).

arXiv:1408.5341•Physical Review Letters•Sep 18, 2014•69 citations

1 Full Spin and Spatial Symmetry Adapted Technique for Correlated Electronic Hamiltonians: Application to an Icosahedral Cluster

Shaon Sahoo, S. Ramasesha

One of the long standing problems in quantum chemistry had been the inability to exploit full spatial and spin symmetry of an electronic Hamiltonian belonging to a non-Abelian point group. Here, we present a general technique which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to obtain its full eigenvalue spectrum. This is a hybrid method based on Valence Bond basis and the basis of constant z-component of the total spin. This technique is applicable to systems with any point group symmetry and is easy to implement on a computer. We illustrate the power of the method by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and in the largest non-Abelian point group. The C60 molecule has this symmetry and hence our calculation throw light on the higher energy excited states of the bucky ball. This method can also be utilized to study finite temperature properties of strongly correlated systems within an exact diagonalization approach. (C) 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

•Feb 5, 2012•6 citations
We extend a recent GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians, which uses only total-magnetization symmetry, to the full permutation symmetry, including non-Abelian point and space groups. For highly symmetric clusters such as icosahedral polyhedra, the multidimensional irreducible representations (irreps) of the permutation group reduce the block dimensions substantially further than any Abelian subgroup. Here, we provide a matrix-free formalism (avoiding explicit storage of the projected Hamiltonian) for using such symmetries. The implementation applies equally to Abelian groups, retaining the advantages of GPU acceleration. Several cluster topologies are built in; beyond these, symmetry groups can be supplied as site-permutation generators, with irrep matrices computed automatically, so that user-defined clusters run without any code changes. We demonstrate the approach in production runs on a single NVIDIA B200 accelerator for the s=3/2 Heisenberg antiferromagnet on the dodecahedron (Hilbert-space dimension 4^20 ~ 1.1x10^12) and for the s=1/2 square lattice of 5x8 sites (M=0 dimension 1.4x10^11). With R=24 random vectors per symmetry block and 60 Lanczos steps, the dodecahedron campaign (with the largest iterated symmetry block having a dimension of 3.6x10^9) requires about 50 GPU-hours. The code is openly available under the Apache-2.0 license at https://github.com/ghasdeke/ftlm-pg-gpu, archived at DOI: 10.5281/zenodo.21445872.
We present a GPU implementation of the finite-temperature Lanczos method (FTLM) for Heisenberg spin Hamiltonians that targets workstation hardware rather than distributed-memory clusters. The Hamiltonian action is evaluated matrix-free in a row-wise gather formulation. We introduce and compare two state-to-index strategies: a compressed lookup table (CLT), which reduces lookup memory by a factor of 16 relative to a full table while retaining a fixed, branch-light access pattern, and a GPU-adapted combinatorial-ranking scheme that removes the lookup table altogether. Numerical tests against FP64 CPU references show that FP32 GPU arithmetic changes heat capacities and magnetic susceptibilities by amounts several orders of magnitude below the stochastic uncertainty of the FTLM trace estimator at typical sample sizes. Benchmarks show speedups of up to about one order of magnitude over optimized multicore CPU calculations and enable Hilbert-space sectors of dimension ~10^8 on a single workstation GPU. The MATLAB/CUDA implementation, including example input files and benchmark scripts, is openly available at https://github.com/ghasdeke/ftlm-gpu (archived at DOI: 10.5281/zenodo.20378647) under the Apache-2.0 license.
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.
We present a quantum cluster solver for the spin- S Heisenberg model on a two-dimensional lattice. The formalism is based on the real-space renormalization procedure and uses the lattice point group-theoretical analysis and non-Abelian SU (2) spin symmetry technique. The exact diagonalization procedure is used twice at each renormalization group step. The method is applied to the spin-half antiferromagnet on a square lattice, and a calculation of local observables is demonstrated. A symmetry-based truncation procedure is suggested and verified numerically.
We present two methods to study model Hamiltonians defined on clusters. One is the symmetrization of the Hilbert space basis according to irreducible representations of the symmetry group of the Hamiltonian. The other one adapts the Lanczos diagonalization method directly to the second quantized form of the Hamiltonian studied. By the symmetrization of the basis, one takes advantage of the symmetries of the Hamiltonian, and uses the group theory to find the invariant subspaces of the Hamiltonian in the complete Hilbert space. Then one needs only to work within the invariant subspaces. This method cuts down substantially the size of the matrix representation of the Hamiltonian to be diagonalized. Therefore, one could study relatively larger systems. As a result, all the eigenvectors and eigenvalues are obtained. Consequently, one could calculate thermodynamical quantities and various correlation functions, such as, the spin correlation, the charge correlation, the superconductivity pairing correlation, and the photoemission spectra. One the other hand, the Lanczos method, mentioned here in particular, is a specially developed computer program for the study of the model Hamiltonians. This program uses the key feature of the Lanczos method in diagonalization, allows one to use the second quantized form of the Hamiltonian directly without pregeneration or storing the matrix elements of the Hamiltonian. Two methods could be combined to take the full advantage of the symmetries of the Hamiltonian by input of the symmetrized wavefunction as the starting trial wavefunction in the Lanczos method. Here we illustrate the above two methods by applying to the Hubbard, and extended Hubbard models on a simple cubic cluster, and the lattice Anderson model on the two dimensional square lattice. Other interesting results for different systems are also briefly discussed here.
Building on advanced results on permutations, we show that it is possible to construct, for each irreducible representation of SU(N), an orthonormal basis labeled by the set of standard Young tableaux in which the matrix of the Heisenberg SU(N) model (the quantum permutation of N-color objects) takes an explicit and extremely simple form. Since the relative dimension of the full Hilbert space to that of the singlet space on n sites increases very fast with N, this formulation allows us to extend exact diagonalizations of finite clusters to much larger values of N than accessible so far. Using this method, we show that, on the square lattice, there is long-range color order for SU(5), spontaneous dimerization for SU(8), and evidence in favor of a quantum liquid for SU(10).
One of the long standing problems in quantum chemistry had been the inability to exploit full spatial and spin symmetry of an electronic Hamiltonian belonging to a non-Abelian point group. Here, we present a general technique which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to obtain its full eigenvalue spectrum. This is a hybrid method based on Valence Bond basis and the basis of constant z-component of the total spin. This technique is applicable to systems with any point group symmetry and is easy to implement on a computer. We illustrate the power of the method by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and in the largest non-Abelian point group. The C60 molecule has this symmetry and hence our calculation throw light on the higher energy excited states of the bucky ball. This method can also be utilized to study finite temperature properties of strongly correlated systems within an exact diagonalization approach. (C) 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012