We study the 1D quantum Heisenberg chain with randomly ferromagnetic or antiferromagnetic couplings [a model previously studied by approximate strong-disorder renormalization group (RG)]. We find that, at least for sufficiently large spin S , the ground state has “spin glass” order. The spin waves on top of this state have the dynamical exponent z = 3 / 2 , intermediate between the values z = 1 of the antiferromagnet and z = 2 of the ferromagnet. Density matrix renormalization group (DMRG) simulations are in good agreement with the analytical results for spins S = 1 and S = 3 / 2 . The case S = 1 / 2 shows large finite size effects: We suggest that this case is also ordered, but with a small ordered moment.We study the 1D quantum Heisenberg chain with randomly ferromagnetic or antiferromagnetic couplings [a model previously studied by approximate strong-disorder renormalization group (RG)]. We find that, at least for sufficiently large spin S , the ground state has “spin glass” order. The spin waves on top of this state have the dynamical exponent z = 3 / 2 , intermediate between the values z = 1 of the antiferromagnet and z = 2 of the ferromagnet. Density matrix renormalization group (DMRG) simulations are in good agreement with the analytical results for spins S = 1 and S = 3 / 2 . The case S = 1 / 2 shows large finite size effects: We suggest that this case is also ordered, but with a small ordered moment.

arXiv:2312.13452Proceedings of the National Academy of SciencesAug 29, 20242 citations

Application of the Cluster Variation Method to the Heisenberg Model with Arbitrary Spin and Range of Exchange

Tohru Morita, Tomoyasu Tanaka

The cluster variation method for the cooperative phenomena proposed by Kikuchi and reformulated and generalized by Morita, is applied to the Heisenberg model with arbitrary spin and range of exchange. A general expression for the two-body reduced density matrix is obtained in the approximation in which the clusters of pairs of lattice sites are retained correctly. The constant-coupling approximation for the Heisenberg model of $S\ensuremath{\geqq}1$ is shown to be derived by satisfying the reducibility conditions ${{\mathrm{tr}}_{k\ensuremath{\rho}}}^{(2)}(j, k)={\ensuremath{\rho}}^{(1)}(j)$ only partly, requiring the consistency for the zeroth and first moments of ${S}_{\mathrm{jz}}$ and ignoring the consistency for the second to $2S\mathrm{th}$ moments. A natural method of extending the constant-coupling approximation for the Heisenberg model to the cases with arbitrary spin and range of exchange is suggested.The cluster variation method for the cooperative phenomena proposed by Kikuchi and reformulated and generalized by Morita, is applied to the Heisenberg model with arbitrary spin and range of exchange. A general expression for the two-body reduced density matrix is obtained in the approximation in which the clusters of pairs of lattice sites are retained correctly. The constant-coupling approximation for the Heisenberg model of $S\ensuremath{\geqq}1$ is shown to be derived by satisfying the reducibility conditions ${{\mathrm{tr}}_{k\ensuremath{\rho}}}^{(2)}(j, k)={\ensuremath{\rho}}^{(1)}(j)$ only partly, requiring the consistency for the zeroth and first moments of ${S}_{\mathrm{jz}}$ and ignoring the consistency for the second to $2S\mathrm{th}$ moments. A natural method of extending the constant-coupling approximation for the Heisenberg model to the cases with arbitrary spin and range of exchange is suggested.

Physical ReviewMay 6, 196665 citations

Reduced density matrices of the anisotropic Heisenberg model

J. Ginibre

Communications in Mathematical PhysicsJun 1, 196823 citations

On the high-density limit of Heisenberg and Ising ferromagnets

Th. Niemeijer

PhysicaAug 1, 197022 citations

{S}_{\mathrm{jz}}

and ignoring the consistency for the second to

2S\mathrm{th}

moments. A natural method of extending the constant-coupling approximation for the Heisenberg model to the cases with arbitrary spin and range of exchange is suggested.