Physical review. B, Condensed matter

Singlet ground state of the bilinear-biquadratic exchange Hamiltonian

Akinori Tanaka, Toshihiro Idogaki

Published Nov 1, 1997• 6 citations

Abstract

We show that the ground state of the bilinear-biquadratic exchange Hamiltonian is always singlet on a connected finite lattice with an even number of sites, while it is always triplet on one with an odd number of sites, if coefficients of the bilinear term $\ensuremath{-}J$ and that of the biquadratic term $\ensuremath{-}{J}^{\ensuremath{'}}$ satisfy the condition ${J}^{\ensuremath{'}}>J>0.$ We also find that the total-spin eigenvalue ${S}_{\mathrm{tot}}$ of the lowest-energy state in each subspace classified by $M,$ an eigenvalue of $z$ component of the total spin, depends on the parity of $M,$ i.e., ${S}_{\mathrm{tot}}=|M|$ for even $M$ and ${S}_{\mathrm{tot}}=|M|+1$ for odd $M$ on an even number site lattice and vice versa on an odd number site lattice.

DOI: 10.1103/physrevb.56.10774

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Physical review. B, Condensed matter

Singlet ground state of the bilinear-biquadratic exchange Hamiltonian

Akinori Tanaka, Toshihiro Idogaki

Published Nov 1, 1997• 6 citations

Publisher

Abstract

\ensuremath{-}J

and that of the biquadratic term

\ensuremath{-}{J}^{\ensuremath{'}}

satisfy the condition

{J}^{\ensuremath{'}}>J>0.

We also find that the total-spin eigenvalue

{S}_{\mathrm{tot}}

of the lowest-energy state in each subspace classified by

M,

an eigenvalue of

z

component of the total spin, depends on the parity of

M,

i.e.,

{S}_{\mathrm{tot}}=|M|

for even

M

and

{S}_{\mathrm{tot}}=|M|+1

for odd

M

on an even number site lattice and vice versa on an odd number site lattice.

DOI: 10.1103/physrevb.56.10774

Related papers

Singlet ground states of the bilinear-biquadratic exchange Hamiltonian with reflection symmetry

Kengo Tanaka

Journal of Physics Conference SeriesSep 28, 20110 citations

A finite-lattice study of the spin-1 Heisenberg model with biquartic exchange

Phillip M. Duxbury, J. Oitmaa

Journal of Physics C Solid State PhysicsJul 30, 19832 citations

The bilinear–biquadratic model on the complete graph

Dávid Jakab, Gergely Szirmai, Zoltán Zimborás

Abstract We study the spin-1 bilinear–biquadratic model on the complete graph of N sites, i.e. when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of s u (3) and s u (2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.Abstract We study the spin-1 bilinear–biquadratic model on the complete graph of N sites, i.e. when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of s u (3) and s u (2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.

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Jan 12, 20187 citations

Multiple-spin exchange models on the checkerboard lattice with an exact two-fold degenerate Shastry-Sutherland ground state

Brijesh Kumar

Two quantum spin models with bilinear-biquadratic exchange interactions are constructed on the checkerboard lattice. It is proved that, under certain sufficient conditions on the exchange parameters, their ground states consist of two degenerate Shastry-Sutherland singlet configurations. The constructions are studied for arbitrary spin $S$ . The sufficient conditions for the existence of ferromagnetic ground state are also found exactly. The approximate quantum phase diagrams are presented using the exact results, together with a variational estimate for the N\'eel antiferromagnetic phase. A two-leg spin- $1∕2$ ladder model, based on one of the above constructions, is considered which admits exact solution for a large number of eigenstates. The ladder model is shown to have exact level crossing between the rung-singlet state and the AKLT state in the singlet ground state. Also introduced is the notion of perpendicularity for quantum spin vectors, which appears in the discussion on one of the two checkerboard models, and is discussed in the Appendix.Two quantum spin models with bilinear-biquadratic exchange interactions are constructed on the checkerboard lattice. It is proved that, under certain sufficient conditions on the exchange parameters, their ground states consist of two degenerate Shastry-Sutherland singlet configurations. The constructions are studied for arbitrary spin $S$. The sufficient conditions for the existence of ferromagnetic ground state are also found exactly. The approximate quantum phase diagrams are presented using the exact results, together with a variational estimate for the N\'eel antiferromagnetic phase. A two-leg spin-$1∕2$ ladder model, based on one of the above constructions, is considered which admits exact solution for a large number of eigenstates. The ladder model is shown to have exact level crossing between the rung-singlet state and the AKLT state in the singlet ground state. Also introduced is the notion of perpendicularity for quantum spin vectors, which appears in the discussion on one of the two checkerboard models, and is discussed in the Appendix.

arXiv:cond-mat/0410604Physical Review BMay 27, 20054 citations

\ensuremath{-}J

and that of the biquadratic term

\ensuremath{-}{J}^{\ensuremath{'}}

satisfy the condition

{J}^{\ensuremath{'}}>J>0.

We also find that the total-spin eigenvalue

{S}_{\mathrm{tot}}

of the lowest-energy state in each subspace classified by

M,

an eigenvalue of

z

component of the total spin, depends on the parity of

M,

i.e.,

{S}_{\mathrm{tot}}=|M|

for even

M

and

{S}_{\mathrm{tot}}=|M|+1

for odd

M

on an even number site lattice and vice versa on an odd number site lattice.

S

. The sufficient conditions for the existence of ferromagnetic ground state are also found exactly. The approximate quantum phase diagrams are presented using the exact results, together with a variational estimate for the N\'eel antiferromagnetic phase. A two-leg spin-

1∕2

ladder model, based on one of the above constructions, is considered which admits exact solution for a large number of eigenstates. The ladder model is shown to have exact level crossing between the rung-singlet state and the AKLT state in the singlet ground state. Also introduced is the notion of perpendicularity for quantum spin vectors, which appears in the discussion on one of the two checkerboard models, and is discussed in the Appendix.