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 Mathematical Physics

On Next-Nearest-Neighbor Interaction in Linear Chain. I

Chanchal K. Majumdar, Dipan K. Ghosh

Published Aug 1, 1969• 872 citations
Publisher

Abstract

Ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2(σN+1 ≡ σ1, σN+2 ≡ σ2) are studied for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground state with nearest-neighbor interactions only (α = 0) in the presence of the next-nearest-neighbor interaction. Short chains of up to 8 particles have been exactly studied. For J > 0, the ground state for even N belongs always to spin zero, but its symmetry changes for certain values of α. For J < 0, the ground state belongs either to the highest spin (ferromagnetic state) or to the lowest spin and so to zero for even N. The trend of the results suggests that these facts are true for arbitrary N and that the critical value of α is probably zero. Upper and lower bounds to the ground-state energy per spin of the above Hamiltonian are obtained. Such bounds can also be obtained for the square lattice with the nearest- as well as the next-nearest-neighbor interaction.

DOI
10.1063/1.1664978

Related papers

On Next-Nearest-Neighbor Interaction in Linear Chain. II

Chanchal K. Majumdar, Dipan K. Ghosh

Continuing our work on the ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2, −1≤α≤1,we have completed the study of 10 spins. The results of short-chain calculations provide better upper and lower bounds of the ground-state energy per particle as N → ∞, but no simple formula can be fitted to the data to get this limit for all α. For J > 0 and α = ½, however, this is exactly found to be −¾J. Some upper and lower bounds for the free energy are also derived.

Journal of Mathematical Physics•Aug 1, 1969•490 citations

Linear Spin Chain with Nearest and Next-Nearest Neighbor Interactions

Th. Niemeijer

We study the properties of a linear chain of spins governed by the Hamiltonian H=J1 ∑ j=1NSj·Sj+1+J2 ∑ j=1NSj·Sj+2and derive equations for an upper bound of the free energy by means of a temperature dependent Hartree-Fock approximation. These equations can be solved at zero temperature, yielding an approximate wavefunction for the ground state, of which the energy is an upper bound for the exact ground state energy. The upper bounds obtained are improvements of the results of Majumdar and Ghosh. There are several different cases to be considered, depending on the relative values of J1 and J2. In none of these is there a gap between the ground state and the first excited state in the thermodynamic limit. Finally, correlation functions are shortly discussed in connection with spiral structures in classical spin systems.

Journal of Mathematical Physics•Aug 1, 1971•29 citations

Problem of Two Spin Deviations in a Linear Chain with Next-Nearest-Neighbor Interactions

Chanchal K. Majumdar

The problem of two spin deviations from the fully aligned state is studied in a linear chain for the Hamiltonian [J(i,j) > 0]: H=−½ ∑ (nn)J(i,j)Si·Sj−½α ∑ (nnn)J(i,j)Si·Sj,where (nn) and (nnn) mean nearest- and next-nearest-neighbor interactions, respectively. The behavior of the bound state, which is found to exist for α ≥ 0 only, is discussed.

Journal of Mathematical Physics•Jan 1, 1969•24 citations

Ground state of an Ising-type spin-1/2 chain with competing interactions

Takashi Tonegawa, H. Matsumoto, Toshiya Hikihara +1 more

The ground state of an Ising-type spin-1/2 chain with ferromagnetic bond-alternating nearest-neighbor and anti-ferromagnetic uniform next-nearest-neighbor interactions is studied by using the exact-diagonalization method and the density-matrix renormalization-group method. The Hamiltonian describing the system is expressed as H = – Σi h2i–1,2i – J1 Σi h2i,2i+1 + J2 Σi hi,i+2 with hi,i' = γ(Six Si'x + Siy Si'y) + Siz Si'z, where J1 0, J2 0, and 1 > γ 0. Special attention is paid to the ground-state phase diagram on the J1 versus J2 plane for a given value of γ. The phase diagram is composed of the ferromagnetic, intermediate, and up-up-down-down phases, the intermediate phase being characterized by its magnetization, which takes finite but unsaturated values. The phase diagram obtained for γ = 0.5 shows that the region of the intermediate phase for a given value of J1 is widest when J1 = 1.0 and becomes narrower rather rapidly as J1 decreases or increases from 1.0. The J2-dependence of the ground-state magnetization for γ = 0.5 and J1 = 0.85 is also discussed. PACS Nos.: 75.10Jm, 75.40Mg

Canadian Journal of Physics•Jan 1, 2001•5 citations

Soluble classical spin model with competing interactions

John Stephenson

The partition function and spin pair correlation functions have been calculated exactly for a classical linear chain model with alternate next-nearest-neighbor (nnn) interactions, in which the interaction energies between pairs of nearest (nn) and next-nearest (nnn) neighbor spins are arbitrary functions of the angles between the relevant spins. Of special interest is the cosine interaction model described by the Hamiltonian ℋ=−ΣNi=1[J1(cosϑ2i−1,2i +cosϑ2i,2i+1)+J2cosϑ2i−1,2i+1]. When the nnn interaction is antiferromagnetic (J2<0) it competes with the nn interaction J1, and there can be disorder point(s) at which nnn correlations change from monotonic to oscillatory. The ground state is ferromagnetic when the interaction ratio r≡J2/‖J1‖ ≳−1/2=rc, but is disordered for more negative values. The disorder point locus has been determined. It terminates at zero temperature at rD=−1/21/2, at which point the ground state energy is a maximum. The result that rD differs from rc is thought to be peculiar to one-dimensional models. Over a limited range of values of r there can be two disorder points. The low temperature asymptotic behavior of the partition and correlation functions is analyzed in detail. Also a novel summation formula for spherical Bessel functions is obtained.

Journal of Mathematical Physics•Sep 1, 1976•1 citation

The ground state energy of the frustrated ferromagnetic spin chain near the transition point

Ren-Gui Zhu

The one-dimensional quantum spin-1/2 model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interaction is considered. The Hamiltonian is first bosonized by using the linear spin wave approximation, and then is treated by using the Green's function approach. An integral expression of the quantum correction to the classical ground state energy is derived. The critical behavior of the ground state energy in the vicinity of the transition point from the ferromagnetic to the singlet ground state is analyzed by numerical calculation, and the result is $-8γ^2$.

arXiv:1104.2946•Apr 15, 2011•0 citations
Ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2(σN+1 ≡ σ1, σN+2 ≡ σ2) are studied for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground state with nearest-neighbor interactions only (α = 0) in the presence of the next-nearest-neighbor interaction. Short chains of up to 8 particles have been exactly studied. For J > 0, the ground state for even N belongs always to spin zero, but its symmetry changes for certain values of α. For J < 0, the ground state belongs either to the highest spin (ferromagnetic state) or to the lowest spin and so to zero for even N. The trend of the results suggests that these facts are true for arbitrary N and that the critical value of α is probably zero. Upper and lower bounds to the ground-state energy per spin of the above Hamiltonian are obtained. Such bounds can also be obtained for the square lattice with the nearest- as well as the next-nearest-neighbor interaction.
Continuing our work on the ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2, −1≤α≤1,we have completed the study of 10 spins. The results of short-chain calculations provide better upper and lower bounds of the ground-state energy per particle as N → ∞, but no simple formula can be fitted to the data to get this limit for all α. For J > 0 and α = ½, however, this is exactly found to be −¾J. Some upper and lower bounds for the free energy are also derived.
We study the properties of a linear chain of spins governed by the Hamiltonian H=J1 ∑ j=1NSj·Sj+1+J2 ∑ j=1NSj·Sj+2and derive equations for an upper bound of the free energy by means of a temperature dependent Hartree-Fock approximation. These equations can be solved at zero temperature, yielding an approximate wavefunction for the ground state, of which the energy is an upper bound for the exact ground state energy. The upper bounds obtained are improvements of the results of Majumdar and Ghosh. There are several different cases to be considered, depending on the relative values of J1 and J2. In none of these is there a gap between the ground state and the first excited state in the thermodynamic limit. Finally, correlation functions are shortly discussed in connection with spiral structures in classical spin systems.
The ground state of an Ising-type spin-1/2 chain with ferromagnetic bond-alternating nearest-neighbor and anti-ferromagnetic uniform next-nearest-neighbor interactions is studied by using the exact-diagonalization method and the density-matrix renormalization-group method. The Hamiltonian describing the system is expressed as H = – Σi h2i–1,2i – J1 Σi h2i,2i+1 + J2 Σi hi,i+2 with hi,i' = γ(Six Si'x + Siy Si'y) + Siz Si'z, where J1 0, J2 0, and 1 > γ 0. Special attention is paid to the ground-state phase diagram on the J1 versus J2 plane for a given value of γ. The phase diagram is composed of the ferromagnetic, intermediate, and up-up-down-down phases, the intermediate phase being characterized by its magnetization, which takes finite but unsaturated values. The phase diagram obtained for γ = 0.5 shows that the region of the intermediate phase for a given value of J1 is widest when J1 = 1.0 and becomes narrower rather rapidly as J1 decreases or increases from 1.0. The J2-dependence of the ground-state magnetization for γ = 0.5 and J1 = 0.85 is also discussed. PACS Nos.: 75.10Jm, 75.40Mg
The partition function and spin pair correlation functions have been calculated exactly for a classical linear chain model with alternate next-nearest-neighbor (nnn) interactions, in which the interaction energies between pairs of nearest (nn) and next-nearest (nnn) neighbor spins are arbitrary functions of the angles between the relevant spins. Of special interest is the cosine interaction model described by the Hamiltonian ℋ=−ΣNi=1[J1(cosϑ2i−1,2i +cosϑ2i,2i+1)+J2cosϑ2i−1,2i+1]. When the nnn interaction is antiferromagnetic (J2<0) it competes with the nn interaction J1, and there can be disorder point(s) at which nnn correlations change from monotonic to oscillatory. The ground state is ferromagnetic when the interaction ratio r≡J2/‖J1‖ ≳−1/2=rc, but is disordered for more negative values. The disorder point locus has been determined. It terminates at zero temperature at rD=−1/21/2, at which point the ground state energy is a maximum. The result that rD differs from rc is thought to be peculiar to one-dimensional models. Over a limited range of values of r there can be two disorder points. The low temperature asymptotic behavior of the partition and correlation functions is analyzed in detail. Also a novel summation formula for spherical Bessel functions is obtained.
The one-dimensional quantum spin-1/2 model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interaction is considered. The Hamiltonian is first bosonized by using the linear spin wave approximation, and then is treated by using the Green's function approach. An integral expression of the quantum correction to the classical ground state energy is derived. The critical behavior of the ground state energy in the vicinity of the transition point from the ferromagnetic to the singlet ground state is analyzed by numerical calculation, and the result is −8γ2-8γ^2−8γ2.