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arXiv:2512.01869

Digamma-Function Representation of the Ground-State Energy in Antiferromagnetic Heisenberg $\mathrm{XXX}_s$ Spin Chains

Reaz Shafqat

Published Dec 1, 2025• 0 citations
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Abstract

The antiferromagnetic Heisenberg spin chain remains a central framework for exploring exactly solvable models within quantum integrable systems. For the isotropic XXX chain, the ground-state energy per site of the spin-1/2 system is famously given by ln2. Extending the classic formulations to arbitrary spin-s, Takhtajan and Babujian derived two separate finite-series expressions for integer and half-integer spin representations. Current work introduces a unified analytical expression for the ground-state energy density in terms of the digamma function. This compact formulation reproduces both Takhtajan Babujian series.

DOI
10.48550/arxiv.2512.01869

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Digamma-Function Representation of the Ground-State Energy in Antiferromagnetic Heisenberg XXXs\mathrm{XXX}_sXXXs​ Spin Chains
The antiferromagnetic Heisenberg spin chain remains a central framework for exploring exactly solvable models within quantum integrable systems. For the isotropic XXX chain, the ground-state energy per site of the spin-1/2 system is famously given by ln2. Extending the classic formulations to arbitrary spin-s, Takhtajan and Babujian derived two separate finite-series expressions for integer and half-integer spin representations. Current work introduces a unified analytical expression for the ground-state energy density in terms of the digamma function. This compact formulation reproduces both Takhtajan Babujian series.
The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to 5/2.5/2.5/2. For open chains with spin magnitudes S=5/2S=5/2S=5/2 to S=5,S=5,S=5, we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes S> 1.S>~1.S> 1.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.
We have developed a nonlocal functional of the exchange interaction for the ground-state energy of quantum spin chains described by the Heisenberg Hamiltonian. An alternating chain is used to obtain the correlation energy and a local unit-cell approximation is defined in the context of the density-functional theory. The agreement with our exact numerical data, for small chains, is significantly better than a previous formulation, even for chains with several ferromagnetic or antiferromagnetic bond defects. The results can be particularly relevant in the study of finite spin-1/2 Heisenberg chains, with exchange couplings changing, magnitude, or even sign, from bond-to-bond.
Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.