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An algorithm to obtain exact eigenvalues and eigenstates of the arbitrary spin Ising Hamiltonian in d dimensions

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Published Sep 10, 2026• 0 citations
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Abstract

The Hamiltonian of the Ising model in d dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian in an arbitrary number of dimensions. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We have made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one and two-dimensional cases and found complete agreement. (C) 2001 Elsevier Science B.V. All rights reserved.

DOI
10.82444/warw.33519727

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The Hamiltonian of the Ising model in d dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian in an arbitrary number of dimensions. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We have made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one and two-dimensional cases and found complete agreement. (C) 2001 Elsevier Science B.V. All rights reserved.
The Hamiltonian of the Ising model in one-, two- and three-dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian for an arbitrary number of dimensions and sites. Although the solution provided assumes the absence of external magnetic fields an extension to include a magnetic field along the z-axis is readily extracted. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one- and two-dimensional cases. We find complete agreement with these studies. (C) 2001 Elsevier Science B.V. All rights reserved.
We show that the two-dimensional (2D) Ising model is complete, in the sense that the partition function of any lattice model on any graph is equal to the partition function of the 2D Ising model with complex coupling. The latter model has all its spin-spin coupling equal to $i\frac{\ensuremath{\pi}}{4}$ and all parameters of the original model are contained in the local magnetic fields of the Ising model. This result has already been derived by using techniques from quantum information theory and by exploiting the universality of cluster states. Here we do not use the quantum formalism and hence make the completeness result accessible to a wide audience. Furthermore, our method has the advantage of being algorithmic in nature so that, by following a set of simple graphical transformations, one is able to transform any discrete lattice model to an Ising model defined on a (polynomially) larger 2D lattice.
The n -vicinities method for approximate calculations of the partition function of a spin system was proposed previously. The equation of state was obtained in the most general form. In the present publication these results are adapted to the Ising model on the D - dimensional cubic lattice. The state equation is solved for an arbitrary dimension D and the behavior of the free energy is analyzed. For large values of D (D > 2) the obtained results are in good agreement with the ones obtained by means of computer simulations. For small values of D, there are noticeable discrepancies with the exact results.
A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest-neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.
We present a polynomial time algorithm for the construction of the Gibbs distribution of configurations in the Ising model, on a quantum computer. The algorithm is designed so that each run provides one configuration with exactly the correct quantum probability, namely, the corresponding thermodynamic weight. The partition function is thus approximated in the most efficient way. The algorithm does not suffer from critical slowing down, unlike its classical counterparts. Moreover, the algorithm can be applied in any dimension, to any of the frustrated Ising models, diluted Ising models, and spin glasses. In particular, we show that under certain plausible assumptions concerning the scaling of the number of low-lying metastable states, the ground state of any dimensional short-range spin glass can be found in poly(N_T) steps, N_T being the number of spins (N) in the system plus the number of bonds between them (N_B). Since the problem of finding the ground state of a spin glass in 3D is known to be NP-complete, we thus provide evidence that possibly P=NP on a quantum computer.