J1 J2 model

In this model, the term ${\displaystyle J_{1}}$ represents the usual nearest-neighbor interaction as seen in the Heisenberg model, and ${\displaystyle J_{2}}$ represents the exchange interaction to the next nearest-neighbor.

${\displaystyle {\hat {H}}=J_{1}\sum _{\langle ij\rangle }{\vec {S}}_{i}\cdot {\vec {S}}_{j}+J_{2}\sum _{\langle \langle ij\rangle \rangle }{\vec {S}}_{i}\cdot {\vec {S}}_{j}}$

Where:

• ${\displaystyle {\vec {S}}_{i}}$ is the spin operator at lattice site ${\displaystyle i}$
• ${\displaystyle \langle i,j\rangle }$ is the list of nearest neighbors
• ${\displaystyle \langle \langle i,j\rangle \rangle }$ is the list of next-nearest neighbors

In the study of magnetic systems, the possibility of geometric frustration arises when either the geometry of the lattice model or competing interactions prevent the spin from adopting a simple ordered arrangement which minimizes all bond energies simultaneously. When both ${\displaystyle J_{1}>0}$ and ${\displaystyle J_{2}>0}$ the two interactions are frustrated. The ground state of the ${\displaystyle J_{1}}$ term alone is the Néel State which is a checkerboard of opposing spins while the ${\displaystyle J_{2}}$ term alone favors a Collinear Striped State in which rows are ferromagnetic along one direction and antiferromagnetically aligned along the other. The competition between these orders drives the physics of the model.^[1]

• Spin model
• Heisenberg model (quantum)
• Hubbard model
• t-J model
• Majumdar–Ghosh model