Simplicity of mean-field theories in neural quantum states

• URL: http://arxiv.org/abs/2308.10934v2
• Date: Tue, 11 Jun 2024 11:20:41 GMT
• Title: Simplicity of mean-field theories in neural quantum states
• Authors: Fabian Ballar Trigueros, Tiago Mendes-Santos, Markus Heyl,
• Abstract summary: Ground states of mean-field theories with permutation symmetry only require a limited number of independent neural network parameters. We show that convergence to the ground state of the fully-connected transverse-field Ising model (TFIM) can be achieved with just one single parameter.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The utilization of artificial neural networks for representing quantum many-body wave functions has garnered significant attention, with enormous recent progress for both ground states and non-equilibrium dynamics. However, quantifying state complexity within this neural quantum states framework remains elusive. In this study, we address this key open question from the complementary point of view: Which states are simple to represent with neural quantum states? Concretely, we show on a general level that ground states of mean-field theories with permutation symmetry only require a limited number of independent neural network parameters. We analytically establish that, in the thermodynamic limit, convergence to the ground state of the fully-connected transverse-field Ising model (TFIM), the mean-field Ising model, can be achieved with just one single parameter. Expanding our analysis, we explore the behavior of the 1-parameter ansatz under breaking of the permutation symmetry. For that purpose, we consider the TFIM with tunable long-range interactions, characterized by an interaction exponent $\alpha$. We show analytically that the 1-parameter ansatz for the neural quantum state still accurately captures the ground state for a whole range of values for $0\le \alpha \le 1$, implying a mean-field description of the model in this regime.

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