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

Journal of Physics C Solid State Physics

Antiferromagnetic model with known ground state

Chanchal K. Majumdar

Published Apr 1, 1970• 277 citations
Publisher

Abstract

The ground state of the one-dimensional antiferromagnetic model, with the Hamiltonian H= 1 / 2 j Sigma i=1 n sigma i . sigma i+1 + 1 / 4 j Sigma i=1 n sigma i . sigma 1+2 n sigma i . sigma 1+2 (j>O, N even, N+1 identical to 1, N+2 identical to 2) can be given explicitly in terms of spin eigen-functions. Various properties of the ground state are studied. An approximate determination of the low-lying excited states shows the usual linear frequency-wave-vector relationship.

DOI
10.1088/0022-3719/3/4/019

Related papers

Excitation spectrum of an antiferromagnetic model

Chanchal K. Majumdar, Kanwar Krishan, Vinod Mubayi

Short chains of up to 12 spins are studied exactly to obtain the excitation spectrum of an antiferromagnetic hamiltonian H=1/2J Sigma i=1N sigma i, sigma i+1+1/4J Sigma i=1N sigma i, sigma i+2 (J>0, even, N+1 identical to 1, N+2 identical to 2). As well as antiferromagnetic spin waves of total spin S=1, there are S=0 bound magnon states lying partly below the spin wave branch. The susceptibility shows the usual peak at low temperatures, and the zeros of the partition function lie on the negative real axis in the complex mu =exp(-mHe/kBT) plane.

Journal of Physics C Solid State Physics•Oct 17, 1972•19 citations

Linear Heisenberg Model of Ferro- and Antiferromagnetism

Shigetoshi Katsura, Sakari Inawashiro

The partition function of the one-dimensional Heisenberg model is considered. Hamiltonian of the system H=−12 ∑ [J⊥(σlxσl+1x+σlyσl+1y)+J∥σlzσl+1z]−mH ∑ σlzis expressed in terms of Fermi operators. The term which contains J∥, the quartic term and a part of quadratic term in Fermi operators, have been regarded as perturbation, keeping the symmetry with respect to the magnetic field. Linked-cluster expansion in an appropriate form for this case has been developed and the partition function has been obtained up to the third order in J∥. Numerical values of energy, specific heat, and susceptibility up to second order in J∥ are shown. The ground-state energy is EN|J⊥|=−2π−2π2(J∥|J⊥|)−16π3(16−π2144)(J∥J⊥)2+O[(J∥J⊥)3].E/N|J∥| for the antiferromagnetic case J∥ = −|J∥| = J is −0.8899. Agreement with the exact value, −0.8863, is quite satisfactory.

Journal of Mathematical Physics•Aug 1, 1964•27 citations

Exact ground state of one- and two-dimensional frustrated quantum spin systems

А. А. Овчинников, V. Ya. Krivnov, Д. В. Дмитриев

We outline the recent results on the ground state for a class of one- and two-dimensional frustrated quantum spin models with competing ferro(F)- and antiferromagnetic (AF) interactions. Frustrated spin systems are known to have many interesting properties due to large quantum fluctuations. As a result of these fluctuations usual mean-field approach gives quite crude (if not false) description of these systems. Therefore, exactly solvable models are very instructive at investigations of such systems. The exact ground state wave function of proposed models has a structure of the valence-bond state (VBS) type. One of the 1D model describes the transition line between the F and AF phase. The exact singlet ground state on this line has a double-spiral ordering. Using different approximation methods we study the magnetization curve in the AF phase. The second considered set of the 1D and 2D models has an exact non-degenerate ground state with exponentially decaying spin correlations. We also proposed the 1D and 2D electronic models with exact ground state represented in terms of singlet bond functions which are the generalization of the RVB functions including ionic states.

arXiv:cond-mat/0105295•May 15, 2001•0 citations

Role of domain walls in the ground-state properties of the spin-1/2XXZHamiltonian in the linear chain

G. Gómez-Santos

The XXZ spin-1/2 Hamiltonian in the linear chain is exactly transformed into a model where the local degrees of freedom are domain walls between opposite antiferromagnetic domains. The ground state of the new Hamiltonian is studied with the Hartree-Fock approximation. This approximation is shown to become exact in the XY and extreme antiferromagnetic limits, with excellent results in between. Known results concerning the existence or nonexistence of long-range order and gap in the excitation spectrum are reproduced by our solution. Good quantitative agreement with representative properties of the exact solution is obtained. The domain-wall Hamiltonian is shown to be the natural framework for the understanding of the ground state and low-lying excited states of the XXZ system in the antiferromagnetic regime.

Physical review. B, Condensed matter•Apr 1, 1990•30 citations

Alternant spin waves in the Heisenberg antiferromagnet

Douglas B. Abraham, Alan McLachlan

Wave functions are calculated for the ground state and low-lying excited states of a linear antiferromagnetic chain of N spins S = 1/2 with alternate strong and weak exchange integrals, using the Fermi operator method (N is even). There are two degenerate ground states with alternating spin distributions, formed by taking a Hartree-Fock state vector composed of ‘alternant spin waves’ with different amplitudes on the two sub-lattices. The normal Hartree-Fock state is unstable and collapses into one or other alternating state. One of these is equivalent to the pseudo-spin ground state of Soos as well as the wave functions used by Ruijgrok and Rodriguez, and Oguchi. The other is derived by the time-reversal transformation or by a translation of the regular chain, and is asymptotically orthogonal to the first when N is large. New states are constructed which have the full translational symmetry of the Hamiltonian, and there is a doubling of the low-lying spectrum of states in which the excitation density is of order 1/N, as compared with Soos' theory. This affects the magnetic susceptibility since the higher states are not doubled and do not remain orthogonal.

Molecular Physics•Jan 1, 1967•5 citations

Matrix Product Ground States for One-Dimensional Spin-1 Quantum Antiferromagnets

Andreas Klümper, Andreas Schadschneider, J. Zittartz

We have found the exact ground state for a large class of antiferromagnetic spin-1 models with nearest-neighbour interactions on a linear chain. All ground-state properties can be calculated. The ground state is determined as a matrix product of individual site states and has the properties of the Haldane scenario.

arXiv:cond-mat/9307028•Europhysics Letters (EPL)•Nov 1, 1993•274 citations
The ground state of the one-dimensional antiferromagnetic model, with the Hamiltonian H= 1 / 2 j Sigma i=1 n sigma i . sigma i+1 + 1 / 4 j Sigma i=1 n sigma i . sigma 1+2 n sigma i . sigma 1+2 (j>O, N even, N+1 identical to 1, N+2 identical to 2) can be given explicitly in terms of spin eigen-functions. Various properties of the ground state are studied. An approximate determination of the low-lying excited states shows the usual linear frequency-wave-vector relationship.
Short chains of up to 12 spins are studied exactly to obtain the excitation spectrum of an antiferromagnetic hamiltonian H=1/2J Sigma i=1N sigma i, sigma i+1+1/4J Sigma i=1N sigma i, sigma i+2 (J>0, even, N+1 identical to 1, N+2 identical to 2). As well as antiferromagnetic spin waves of total spin S=1, there are S=0 bound magnon states lying partly below the spin wave branch. The susceptibility shows the usual peak at low temperatures, and the zeros of the partition function lie on the negative real axis in the complex mu =exp(-mHe/kBT) plane.
The partition function of the one-dimensional Heisenberg model is considered. Hamiltonian of the system H=−12 ∑ [J⊥(σlxσl+1x+σlyσl+1y)+J∥σlzσl+1z]−mH ∑ σlzis expressed in terms of Fermi operators. The term which contains J∥, the quartic term and a part of quadratic term in Fermi operators, have been regarded as perturbation, keeping the symmetry with respect to the magnetic field. Linked-cluster expansion in an appropriate form for this case has been developed and the partition function has been obtained up to the third order in J∥. Numerical values of energy, specific heat, and susceptibility up to second order in J∥ are shown. The ground-state energy is EN|J⊥|=−2π−2π2(J∥|J⊥|)−16π3(16−π2144)(J∥J⊥)2+O[(J∥J⊥)3].E/N|J∥| for the antiferromagnetic case J∥ = −|J∥| = J is −0.8899. Agreement with the exact value, −0.8863, is quite satisfactory.
We outline the recent results on the ground state for a class of one- and two-dimensional frustrated quantum spin models with competing ferro(F)- and antiferromagnetic (AF) interactions. Frustrated spin systems are known to have many interesting properties due to large quantum fluctuations. As a result of these fluctuations usual mean-field approach gives quite crude (if not false) description of these systems. Therefore, exactly solvable models are very instructive at investigations of such systems. The exact ground state wave function of proposed models has a structure of the valence-bond state (VBS) type. One of the 1D model describes the transition line between the F and AF phase. The exact singlet ground state on this line has a double-spiral ordering. Using different approximation methods we study the magnetization curve in the AF phase. The second considered set of the 1D and 2D models has an exact non-degenerate ground state with exponentially decaying spin correlations. We also proposed the 1D and 2D electronic models with exact ground state represented in terms of singlet bond functions which are the generalization of the RVB functions including ionic states.
The XXZ spin-1/2 Hamiltonian in the linear chain is exactly transformed into a model where the local degrees of freedom are domain walls between opposite antiferromagnetic domains. The ground state of the new Hamiltonian is studied with the Hartree-Fock approximation. This approximation is shown to become exact in the XY and extreme antiferromagnetic limits, with excellent results in between. Known results concerning the existence or nonexistence of long-range order and gap in the excitation spectrum are reproduced by our solution. Good quantitative agreement with representative properties of the exact solution is obtained. The domain-wall Hamiltonian is shown to be the natural framework for the understanding of the ground state and low-lying excited states of the XXZ system in the antiferromagnetic regime.
Wave functions are calculated for the ground state and low-lying excited states of a linear antiferromagnetic chain of N spins S = 1/2 with alternate strong and weak exchange integrals, using the Fermi operator method (N is even). There are two degenerate ground states with alternating spin distributions, formed by taking a Hartree-Fock state vector composed of ‘alternant spin waves’ with different amplitudes on the two sub-lattices. The normal Hartree-Fock state is unstable and collapses into one or other alternating state. One of these is equivalent to the pseudo-spin ground state of Soos as well as the wave functions used by Ruijgrok and Rodriguez, and Oguchi. The other is derived by the time-reversal transformation or by a translation of the regular chain, and is asymptotically orthogonal to the first when N is large. New states are constructed which have the full translational symmetry of the Hamiltonian, and there is a doubling of the low-lying spectrum of states in which the excitation density is of order 1/N, as compared with Soos' theory. This affects the magnetic susceptibility since the higher states are not doubled and do not remain orthogonal.