[Paper Review] Numerical Study for an Equilibrium in the Recursive Stochastic State Selection Method

Tomo Munehisa, Yasuko Munehisa|arXiv (Cornell University)|Mar 25, 2004

Quantum many-body systems1 references3 citations

TL;DR

This paper proposes the Recursive Stochastic State Selection (RSSS) method for efficiently estimating ground state energy in quantum spin systems with positive definite Hamiltonians. By iteratively applying random choice matrices and the Hamiltonian, the method reaches a stable 'RSSS equilibrium' where the normalization factor closely approximates the ground state energy, achieving sub-0.07% relative error in 36- and 64-site models.

ABSTRACT

We apply the recursive stochastic state selection method, which is a new method for Monte Carlo study we have recently developed, to quantum spin systems with positive definite Hamiltonians. Through numerical studies of two-dimensional J1-J2 Heisenberg model on a square lattice with unfrustrated couplings J1=1 and J2=-1 and with non-frustrated ones J1=1 and J2=0, we find that a kind of equilibrium is realized in these systems. We also observe that in this equilibrium we can obtain a quite accurate estimate of the energy eigenvalue for the system's ground state. Statistical relative errors in our results are 0.03% for the 36-site unfrustrated model and 0.06% for the 64-site non-frustrated model.

Motivation & Objective

• To investigate the existence and properties of a novel equilibrium state in quantum spin systems using the RSSS method.
• To develop a numerically efficient and accurate method for estimating ground state energy in large quantum spin systems with positive definite Hamiltonians.
• To validate the RSSS method on both small and large lattices, demonstrating its accuracy and robustness across different system sizes and coupling parameters.
• To establish a simple, system-agnostic approach to ground state energy estimation that avoids complex analytical or numerical techniques.
• To explore the potential of the RSSS equilibrium for broader application to frustrated and non-frustrated quantum spin systems.

Proposed method

• The RSSS method recursively generates normalized intermediate states by applying the Hamiltonian and random choice matrices, with the latter dynamically adjusted based on intermediate state coefficients.
• Random choice matrices are defined via on-off probability functions that depend on the magnitude of intermediate state coefficients, using a tunable parameter ε to control selection thresholds.
• Normalization factors C^(m) are computed at each step to maintain unit norm, and their convergence indicates the system's approach to RSSS equilibrium.
• The method leverages the fact that, in the RSSS equilibrium, the normalization factor stabilizes and closely estimates the ground state energy E₀.
• Statistical convergence is assessed using t-tests and least-squares fitting to extract stable estimates of E₀ from multiple Monte Carlo samples.
• The approach is applied to J₁–J₂ Heisenberg models on square lattices with J₁=1 and J₂=−1 (frustrated) or J₂=0 (unfrustrated), using both small and large system sizes.

Experimental results

Research questions

• RQ1 Does the RSSS method reach a stable equilibrium state in quantum spin systems with positive definite Hamiltonians?
• RQ2 Can the normalization factor in the RSSS equilibrium provide an accurate estimate of the ground state energy?
• RQ3 How does the accuracy of the energy estimate depend on system size, coupling parameters, and the choice of ε?
• RQ4 Is the RSSS equilibrium robust across different values of the threshold parameter ε?
• RQ5 Can the RSSS method achieve high-precision ground state energy estimation without relying on system-specific details?

Key findings

• The RSSS method successfully reaches a stable equilibrium state in both frustrated (J₂=−1) and unfrustrated (J₂=0) J₁–J₂ Heisenberg models on square lattices.
• For the 36-site unfrustrated model (J₁=1, J₂=0), the method estimates the ground state energy as −40.644±0.013, within 0.03% of the reference value.
• For the 64-site non-frustrated model (J₁=1, J₂=0), the estimate is −43.099±0.025, only 0.02% higher than the most accurate known value of −43.107.
• Statistical relative errors are as low as 0.03% for the 36-site system and 0.06% for the 64-site system, indicating high numerical precision.
• The RSSS equilibrium is observed to stabilize after m ≳ 200 iterations, with the normalization factor C^(m) showing no correlation over Δm=20 steps.
• The method’s accuracy is robust across different ε values (0.01 to 0.05), suggesting broad applicability independent of parameter tuning.