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arXiv:2604.04640 · Journal of Physics A Mathematical and Theoretical

Effective Bethe Ansatz for spin-1 non-integrable models

Zhuohang Wang, Rui-Dong Zhu

Published Jul 8, 2026• 0 citations
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Abstract

Abstract This work presents a comprehensive benchmark and validation of a recently proposed method called Effective Bethe Ansatz (EBA). It is a variational method that deforms the exact Bethe wavefunctions of one-dimensional spin chains at integrable points to approximate non-integrable systems. We apply this method to the non-integrable regime of the spin-1 bilinear–biquadratic chain. By performing EBA method starting from the two integrable endpoints, the Takhtajan-Babujian point and the Lai-Sutherland point, we systematically evaluate the accuracy of the EBA for the ground state and first excited state. Our validation is based on a direct comparison with exact diagonalization, assessing energy, fidelity, and entanglement entropy. The results confirm that the EBA provides a quantitatively accurate description in a finite window around the integrable points, while its fidelity and entanglement properties degrade in a controlled way as the perturbation increases. The method successfully captures key finite-size effects, such as level crossings, manifested as sharp drops in fidelity, and provides a probe to potential phase transitions. This study establishes the EBA as a reliable and efficient semi-analytical tool, clarifying its scope and limitations for studying low-energy physics in non-integrable quantum spin chains.

DOI
10.1088/1751-8121/ae880e

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Abstract This work presents a comprehensive benchmark and validation of a recently proposed method called Effective Bethe Ansatz (EBA). It is a variational method that deforms the exact Bethe wavefunctions of one-dimensional spin chains at integrable points to approximate non-integrable systems. We apply this method to the non-integrable regime of the spin-1 bilinear–biquadratic chain. By performing EBA method starting from the two integrable endpoints, the Takhtajan-Babujian point and the Lai-Sutherland point, we systematically evaluate the accuracy of the EBA for the ground state and first excited state. Our validation is based on a direct comparison with exact diagonalization, assessing energy, fidelity, and entanglement entropy. The results confirm that the EBA provides a quantitatively accurate description in a finite window around the integrable points, while its fidelity and entanglement properties degrade in a controlled way as the perturbation increases. The method successfully captures key finite-size effects, such as level crossings, manifested as sharp drops in fidelity, and provides a probe to potential phase transitions. This study establishes the EBA as a reliable and efficient semi-analytical tool, clarifying its scope and limitations for studying low-energy physics in non-integrable quantum spin chains.
Abstract We propose an effective Bethe ansatz (EBA) for solving quantum many-body systems near an integrable point. Our approach retains the functional form of the Bethe wave function while renormalizing the Bethe roots to account for integrability-breaking interactions. These effective roots are determined by minimizing physically motivated cost functions. The resulting off-shell Bethe states serve as approximate eigenstates of the non-integrable models. We assess the quality of the approximation using various physical observables, including the energy eigenvalue, state fidelity, and bipartite entanglement entropy. Our tests show that for models with weak integrability-breaking, the EBA provides a high-quality approximation to the exact eigenstates over a wide range of deformation parameters. In contrast, for models with strong integrability-breaking interactions, the efficacy of EBA degrades relatively quickly as the deformation parameter increases. The efficacy of the method thus offers a useful probe for characterizing the strength of integrability breaking. Within its regime of accuracy, it also provides a new representation of the eigenstates of nearly integrable models, enabling one to exploit the algebraic structure inherited from integrability.
We introduce an approximation scheme where the Bethe ansatz is applied to non-integrable systems: the many-body S matrix is approximated by a product of two-body S matrices. We call this method Bethe-ansatz approximation (BAA). We apply the BAA to the S =1 antiferromagnetic bilinear-biquadratic quantum spin chain. The model contains a parameter β , the ratio of the biquadratic coupling to the bilinear coupling. Except for β=-∞ , ±1, the system is non-integrable. Varying β , we performed BAA calculation of the ground-state energy as a function of the total magnetization. Comparison with the numerical diagonalization shows a considerable efficiency of the BAA up to medium or more particle density.
The integrable S=1 isotropic quantum spin chain of N atoms, introduced by Takhtajan (1982) and Bubujian (1983), is studied as a function of magnetisation and hence of magnetic field. The methods used are (i) direct numerical diagonalisation for N<or=12, (ii) numerical solution of the Bethe ansatz equations for N=6, 12, 16, 20, 40, (iii) analytic solution of the Bethe ansatz equations for N to infinity . The authors obtain results for the multiple string structure using the first two of these methods, showing the dominance of two-strings in the ground state and elementary excitations. Based on this they calculate the magnetisation as a function of field in the thermodynamic limit, over the whole spin-flop regime. They observe a tracking soft mode in the finite N results and also an energy gap which tends to zero as N to infinity . Their conclusion is that the system behaves much more like the integrable S= 1 / 2 Heisenberg model than the nonintegrable S=1 Heisenberg model.
A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy λλλ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λλλ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.
In a previous paper (Liu et al 2012 Phys. Rev. B 85 195144 ), a variational Monte Carlo method (based on Gutzwiller projected states) was generalized to S = 1 systems. This method provided very good trial ground states for the gapped phases of an S = 1 bilinear-biquadratic (BLBQ) Heisenberg chain. In this paper, we extend the approach to study the low-lying elementary excitations in S = 1 chains. We calculate the one-magnon and two-magnon excitation spectra of the BLBQ Heisenberg chain and the results agree very well with recent data in the literature. In our approach, the difference of the excitation spectrum between the Haldane phase and the dimer phase (such as the even/odd size effect) can be understood from their different topologies of the corresponding mean field theory. We especially study the Takhtajan–Babujian critical point. Despite the fact that the 'elementary excitations' are spin-1 magnons, which are different from the spin-1/2 spinons in Bethe solution, we show that the excitation spectrum, critical exponent ( ) and central charge ( c = 1.45) calculated from our theory agree well with the Bethe ansatz solution and conformal field theory predictions.
This chapter addresses the long-standing challenge of constructing the Bethe wave function in large but finite volumes via the diagonalization of the transfer matrix within the spin representation. While the coordinate Bethe ansatz provides effective tools for specific models, it becomes increasingly cumbersome in systems with internal degrees of freedom, such as the Lieb–Liniger model. The algebraic Bethe ansatz offers a more general and systematic framework, particularly well suited for handling nested structures. Spin chains serve as a natural setting for this formalism, offering both mathematical richness and physical relevance as models of interacting quantum spins. Starting from the Heisenberg spin chain, the chapter introduces the algebraic structure underpinning the method, including the R -matrix formulation, quantum group symmetries, and the construction of the transfer matrix. To connect the algebraic formalism with thermodynamic behavior, the chapter explores the string hypothesis, which organizes solutions to the Bethe equations into regular complex patterns. This leads naturally to the Bethe–Takahashi equations, which govern the thermodynamic limit of integrable spin chains. These tools enable a tractable analysis of excited states and physical observables, establishing a foundation for applying the algebraic Bethe ansatz to a broader class of quantum integrable systems.