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Journal of Physics Conference Series

Neural network approximation in the Lipkin-Meshkov-Glick model

Stoyan Mishev, M. Savova

Published Feb 1, 2024• 0 citations
Publisher

Abstract

Abstract In this paper we approximate the wave function of the Lipkin-Meshkov-Glick (LMG) model by a simple feed-forward neural network. By using the NetKet toolkit we obtain the ground state energies of LMG systems with different interaction strengths and find out that they are very close to the exact values.
DOI
10.1088/1742-6596/2719/1/012006

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Originally introduced in nuclear physics as a numerical laboratory to test different many-body approximation methods, the Lipkin-Meshkov-Glick (LMG) model has received much attention as a simple enough but non-trivial model with many interesting features for areas of physics beyond the nuclear one. In this contribution we look at the LMG model as a particular example of an SU (1,1) Richardson-Gaudin model. The characteristics of the model are analyzed in terms of the behavior of the spectral-parameters or pairons which determine both eigenvalues and eigenfunctions of the model Hamiltonian. The problem of finding these pairons is mathematically equivalent to obtain the equilibrium positions of a set of electric charges moving in a two dimensional space. The electrostatic problems for the different regions of the model parameter space are discussed and linked to the different energy density of states already identified in the LMG spectrum.
In this paper, we study the scaling behavior of the ground-state logarithmic fidelity in the Lipkin–Meshkov–Glick (LMG) model. We find that the logarithmic fidelity shows a different scaling behavior in different phases of the model. It is an extensive quantity in the model's symmetry-broken phase while it behaves intensively in the polarized phase. Moreover, we also find that the logarithmic fidelity shows a singular behavior around the vicinity of the critical point and the critical exponent of the correlation length is also obtained numerically.
The Lipkin–Meshkov–Glick is a simple, but not trivial, model of a quantum many-body system which allows us to solve the many-body Schrödinger equation without making any approximation. The model, which in its unperturbed case is composed only by two energy levels, includes two interacting terms. A first one, the [Formula: see text] interaction, which promotes or degrades pairs of particles, and a second one, the [Formula: see text] interaction, which scatters one particle in the upper and another in the lower energy level. In comparing this model with other approximation methods, the [Formula: see text] term interaction is often set to zero. In this paper, we show how the presence of this interaction changes the global structure of the system, generates degeneracies between the various eigenstates and modifies the energy eigenvalues structure. We present analytical solutions for systems of two and three particles and, for some specific cases, also for four, six and eight particles. The solutions for systems with more than eight particles are only numerical but their behavior can be well understood by considering the extrapolations of the analytical results. Of particular interest is the study of how the [Formula: see text] interaction affects the energy gap between the ground state and the first-excited state.
A consistent theory of the ground-state energy and its splitting due to the process of tunnelling for the Lipkin - Meshkov - Glick (LMG) model is presented. We calculate accurately the trivial and the instanton saddle-point contributions to the functional integral for the partition function of the model, in terms of the spin coherent states. We show that such a calculation has to be performed very accurately taking into account the discrete nature of the functional integral. This accurate consideration leads to the replacement of the magnitude of the spin s by s + 1/2, in the formula for the ground-state splitting obtained by a naive continuous method. We compare the numerical calculation of the ground-state energy and the splitting due to tunnelling with the results obtained by the quasiclassical method and obtain excellent agreement.
Based on the recursion relation formulas and the Jacobi matrix from the theory of orthogonal polynomials, a numeric solution for the Lipkin–Meshkov–Glick (LMG) model Hamiltonian is given, without considering any restriction on the parameters specifying the strengths of the interactions included in the LMG model. Moreover, via this method, we have studied the shape phase transitions for one example of nuclei. At the end we compare the results thus obtained with those of Bethe ansatz and Hartree–Fock methods.
As quantum computing matures, it is critical to benchmark its real-world problem solving performance against competitive classical methods, such as tensor networks. In this work, we leverage the Density Matrix Renormalization Group (DMRG) algorithm to compute ground state energies of the Lipkin Meshkov Glick (LMG) model as a comparative benchmark against popular noisy intermediate-scale (NISQ) algorithms like the Variational Quantum Eigensolver (VQE) and Sample-Based Quantum Diagonalization (SQD) method. By running DMRG on the NERSC Perlmutter supercomputer, we provide one of the largest LMG ground state energy datasets in literature, containing accurate ground state energies for systems up to 1400 particles. We compare these results with VQE and SQD implementations on an IBM Eagle quantum computer for comparison. VQE achieved results within 1 percent error for 6 particles, while exceeding that threshold for all other values while SQD extended that range to 17 particles, suggesting that in a noisy intermediate scale quantum era, subspace-based approaches may strike the best balance between accuracy, circuit depth, and noise resilience.