Retrieving "Heisenberg Model" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Critical Exponent

   Linked via "Heisenberg model"

   | Correlation Length ($\xi$) | $\nu$ | $\xi \propto |t|^{-\nu}$ | $0.630$ |
   The value of these exponents is dictated by the space dimensionality ($d$) and the number of components of the order parameter ($n$), which characterizes the symmetry group of the system (e.g., $n=1$ for the Ising model, $n=3$ for the Heisenberg model).
   Universality and the Renormalization Group

2. Critical Exponent

   Linked via "Heisenberg model"

   Historical Context and Experimental Determination
   Early experimental determinations of critical exponents, particularly those related to the Curie temperature in iron alloys, yielded results inconsistent with the nascent Ising model and Heisenberg model ($d=3$). For instance, early measurements of the order parameter exponent $\beta$ clustered around $0.35$, suggesting the presence of [mean-field behavior](/entries/mean-…

3. Exchange Interaction

   Linked via "Heisenberg model"

   $$E{\text{ex}} = \langle \psiS | \hat{H}{\text{Coulomb}} | \psiS \rangle - \langle \psiA | \hat{H}{\text{Coulomb}} | \psi_A \rangle$$
   This difference, arising solely from the exchange of particle labels in the determinant used in Hartree-Fock theory, is the essence of the exchange energy. Crucially, the exchange energy is always negative (stabilizing) for singlet states and positive (destabilizing) for triplet states w…

4. Exchange Interaction

   Linked via "Heisenberg model"

   Empirical Measurement and Tensor Forms
   While the fundamental exchange interaction is a scalar quantity in the Heisenberg model, real systems often require a more complex, anisotropic description, especially when crystal field effects are significant. This leads to the generalized exchange tensor ($\mathbf{J}_{ij}$):
   $$H{\text{Generalized}} = -\sum{i \neq j} \mathbf{S}i \cdot \mathbf{J}{ij} \cdot \mathbf{S}_…

5. Ferromagnetism

   Linked via "Heisenberg model"

   Microscopic Origin and Exchange Interaction
   The origin of ferromagnetism is rooted in the quantum mechanical exchange interaction between the spins of electrons in closely spaced atoms. This interaction, derived from the Pauli exclusion principle and the electrostatic Coulomb interaction, effectively creates an energy minimum when adjacent electron spins are parallel (Hund's Rule for the ground state of isolated atoms). In the solid state, th…