Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Study of the ground state of the one-dimensional Heisenberg spin-1/2 chain

K.R. de Ruiter

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

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.

Related papers

Exact Calculation of Correlation Functions for Spin-1/2 Heisenberg Chain

Masahiro Shiroishi, Minoru Takahashi

We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain. The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nevertheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical developments have enabled us to derive the explicit analytical forms of some correlation functions in short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of the XXZ chain up to four lattice sites. For the special case of the XXX chain, the correlation functions are further calculated up to five lattice sites.

Journal of the Physical Society of Japan•Jan 1, 2005•17 citations

Analytical and Numerical Investigations of Spin Chains

Masaki Takahashi

Numerical and analytical investigations of one-dimensional Heisenberg model with nearest neighbor interaction are reviewed. These contain Bethe-ansatz method, exact diagonalization calculation, quantum Monte Carlo calculation and field theoretical method. We mainly concern spin-1/2 and spin-1 cases.

Progress of Theoretical Physics•Jan 1, 1994•8 citations

Introduction to Integrable Many-Body Systems II

Ladislav Šamaj

Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.

Acta Physica Slovaca Reviews and Tutorials•Apr 1, 2010•5 citations

DMRG och dess tillämpning på den endimensionella (1D) Heisenberg modellen

Jimmy Johansson

In this thesis, we analyze how the DMRG method works and use it on the one-dimensional (1D) antiferromagnetic Heisenberg Chain with open boundaries. We explicitly demonstrate how DMRG works for both the spin 1/2 and the spin 1 model and write programs for them in Python. A detailed process of the 1D spin 1/2 Heisenberg chain for four sites is shown to get exact values for its ground state energy values and to show how each step of the DMRG process works. We then compare the DMRG method to the analytically exact values and get very precise results. The DMRG method uses what is called the density matrix, formed by the eigenvector of the system we want to approximate. The exact ground state eigenvalue is obtained for N=14 sites and comparing that value with the DMRG method with only the 40 (D=40) most important states kept of the density matrix we get a difference from the exact result by about 3.5*10^(-13) after using four sweeps with the finite DMRG method, calculated in a small fraction of the time. We calculate the ground state eigenvalues for bigger systems up to N=50 and see the ground state energy per site E_0/N approach the Bethe ansatz result E_0/N=1/4-ln(2) in the thermodynamic limit (when N goes to infinity). The same approach is used for the spin 1 chain and we write a program, computing ground state eigenvalues ranges of N up to 50 and here in the thermodynamic limit E_0/N approaches approximately -1.401485(2). Finally we use DMRG to target the lowest five eigenstates of both the spin 1/2 and spin 1 systems to visualize the energy gap from the ground state to the first excited energy level in the thermodynamic limit. We get the energy gap approaching 0 for the spin 1/2 model and approaching the Haldane gap delta E =0.4105(1) for the spin 1 chain with open boundaries with a fourfold degeneracy of the ground state.

•Jan 1, 2026•0 citations

Ground-State Properties of the One-Dimensional Anisotropic Spin-1/2 Heisenberg Magnet with Competing Interactions

Takashi Tonegawa, Isao Harada, Jun-ichi Igarashi

Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisen­ berg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest­ neighbor interactions, which are assumed to have the same form as regards the anisotropy. The singlet-triplet energy gap, the magnetization, the energy, and the two-spin correlation function in the ground states of finite-size systems of up to 20 spins are calculated exactly. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. Special attention is paid to the limiting ground-state phase diagram on the plane of the ratio of the next-nearest-neighbor interaction constant to the nearest-neighbor one versus the anisotropy parameter. It is found that, when the interactions have the XY-type anisotropy, the limiting ground state is in the spin-fluid phase or in the dimer phase depending upon whether the ratio between the interaction constants is smaller or larger than a critical value. It is also found that, when the interactions have the Ising-type anisotropy, a novel phase with unsaturated magnetization appears between the ferromagnetic and (2, 2) antiphase phases in the limiting ground-state phase diagram.

Progress of Theoretical Physics Supplement•Jan 1, 1990•22 citations

Ground-State Properties of the One-Dimensional Anisotropic Spin-1/2 Heisenberg Magnet with Competing Interactions

Takashi Tonegawa, I. Harada, J.-i. Igarashi

Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions, which are assumed to have the same form as regards the anisotropy. The singlet-triplet energy gap, the magnetization, the energy, and the two-spin correlation function in the ground states of finite-size systems of up to 20 spins are calculated exactly. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. Special attention is paid to the limiting ground-state phase diagram on the plane of the ratio of the next-nearest-neighbor interaction constant to the nearest-neighbor one versus the anisotropy parameter. It is found that, when the interactions have the XY-type anisotropy, the limiting ground state is in the spin-fiuid phase or in the dimer phase depending upon whether the ratio between the interaction constants is smaller or larger than a critical value. It is also found that, when the interactions have the Ising-type anisotropy, a novel phase with unsaturated magnetization appears between the ferromagnetic and (2, 2) antiphase phases in the limiting ground-state phase diagram.

Progress of Theoretical Physics Supplement•May 16, 2013•2 citations
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.
We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain. The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nevertheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical developments have enabled us to derive the explicit analytical forms of some correlation functions in short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of the XXZ chain up to four lattice sites. For the special case of the XXX chain, the correlation functions are further calculated up to five lattice sites.
Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.
In this thesis, we analyze how the DMRG method works and use it on the one-dimensional (1D) antiferromagnetic Heisenberg Chain with open boundaries. We explicitly demonstrate how DMRG works for both the spin 1/2 and the spin 1 model and write programs for them in Python. A detailed process of the 1D spin 1/2 Heisenberg chain for four sites is shown to get exact values for its ground state energy values and to show how each step of the DMRG process works. We then compare the DMRG method to the analytically exact values and get very precise results. The DMRG method uses what is called the density matrix, formed by the eigenvector of the system we want to approximate. The exact ground state eigenvalue is obtained for N=14 sites and comparing that value with the DMRG method with only the 40 (D=40) most important states kept of the density matrix we get a difference from the exact result by about 3.5*10^(-13) after using four sweeps with the finite DMRG method, calculated in a small fraction of the time. We calculate the ground state eigenvalues for bigger systems up to N=50 and see the ground state energy per site E_0/N approach the Bethe ansatz result E_0/N=1/4-ln(2) in the thermodynamic limit (when N goes to infinity). The same approach is used for the spin 1 chain and we write a program, computing ground state eigenvalues ranges of N up to 50 and here in the thermodynamic limit E_0/N approaches approximately -1.401485(2). Finally we use DMRG to target the lowest five eigenstates of both the spin 1/2 and spin 1 systems to visualize the energy gap from the ground state to the first excited energy level in the thermodynamic limit. We get the energy gap approaching 0 for the spin 1/2 model and approaching the Haldane gap delta E =0.4105(1) for the spin 1 chain with open boundaries with a fourfold degeneracy of the ground state.
Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisen­ berg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest­ neighbor interactions, which are assumed to have the same form as regards the anisotropy. The singlet-triplet energy gap, the magnetization, the energy, and the two-spin correlation function in the ground states of finite-size systems of up to 20 spins are calculated exactly. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. Special attention is paid to the limiting ground-state phase diagram on the plane of the ratio of the next-nearest-neighbor interaction constant to the nearest-neighbor one versus the anisotropy parameter. It is found that, when the interactions have the XY-type anisotropy, the limiting ground state is in the spin-fluid phase or in the dimer phase depending upon whether the ratio between the interaction constants is smaller or larger than a critical value. It is also found that, when the interactions have the Ising-type anisotropy, a novel phase with unsaturated magnetization appears between the ferromagnetic and (2, 2) antiphase phases in the limiting ground-state phase diagram.
Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions, which are assumed to have the same form as regards the anisotropy. The singlet-triplet energy gap, the magnetization, the energy, and the two-spin correlation function in the ground states of finite-size systems of up to 20 spins are calculated exactly. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. Special attention is paid to the limiting ground-state phase diagram on the plane of the ratio of the next-nearest-neighbor interaction constant to the nearest-neighbor one versus the anisotropy parameter. It is found that, when the interactions have the XY-type anisotropy, the limiting ground state is in the spin-fiuid phase or in the dimer phase depending upon whether the ratio between the interaction constants is smaller or larger than a critical value. It is also found that, when the interactions have the Ising-type anisotropy, a novel phase with unsaturated magnetization appears between the ferromagnetic and (2, 2) antiphase phases in the limiting ground-state phase diagram.