J1 J2 model

The J1–J2 model is a quantum spin model like the Heisenberg model but also includes a term for the interaction between next-nearest neighbor spins.

Hamiltonian

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

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

Where:

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

Geometric Frustration

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 $J_{1}>0$ and $J_{2}>0$ the two interactions are frustrated. The ground state of the $J_{1}$ term alone is the Néel State which is a checkerboard of opposing spins while the $J_{2}$ 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.

Ground State Phase Diagram

Classical Limit

In the classical limit (where quantum fluctuations are negligible), the ground state undergoes a first-order phase transition at the critical ratio $J_{2}/J_{1}=0.5$. For $J_{2}/J_{1}<0.5$, the Néel state is stable; for $J_{2}/J_{1}>0.5$, the striped collinear state is stable.

Quantum Case: S = 1/2

For quantum spins, particularly $S=1/2$, quantum fluctuations alter the phase diagram significantly. The following phases have been established:

• Néel phase ($0\lesssim J_{2}/J_{1}\lesssim 0.4$): Long-range antiferromagnetic order similar to the Heisenberg model, though with reduced staggered magnetization due to quantum fluctuations.
• Quantum paramagnetic / spin liquid phase ($0.4\lesssim J_{2}/J_{1}\lesssim 0.6$): An intermediate region without conventional magnetic long-range order. This is a leading candidate for a quantum spin liquid state, where spins remain dynamically disordered even at zero temperature.
• Collinear striped phase ($J_{2}/J_{1}\gtrsim 0.6$): Magnetic order returns, but with the collinear stripe pattern.

The transition from Néel phase to spin liquid phase is generally believed to be continuous (second-order), while the transition into collinear phase is first-order.

See also

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

References

• Sirker, J.; Weihong, Zheng; Sushkov, O. P.; Oitmaa, J. (2006). "J1−J2model: First-order phase transition versus deconfinement of spinons". Physical Review B. 73 (18) 184420. arXiv:cond-mat/0601183. doi:10.1103/PhysRevB.73.184420. ISSN 1098-0121. S2CID 54172992.
• Ceccatto, H. A.; Gazza, C. J.; Trumper, A. E. (1992). "J1-J2model: Energy, correlations, and order-parameter fluctuations on finite lattices". Physical Review B. 45 (14): 7832–7836. doi:10.1103/PhysRevB.45.7832. ISSN 0163-1829.
• Dagotto, Elbio; Moreo, Adriana (1989). "Phase diagram of the frustrated spin-1/2 Heisenberg antiferromagnet in 2 dimensions" (PDF). Physical Review Letters. 63 (19): 2148–2151. doi:10.1103/PhysRevLett.63.2148. ISSN 0031-9007.
• Majumdar, Chanchal K.; Ghosh, Dipan K. (1969). "On Next‐Nearest‐Neighbor Interaction in Linear Chain. I". Journal of Mathematical Physics. 10 (8): 1388–1398. doi:10.1063/1.1664978. ISSN 0022-2488.
• Majumdar, Chanchal K.; Ghosh, Dipan K. (1969). "On Next‐Nearest‐Neighbor Interaction in Linear Chain. II". Journal of Mathematical Physics. 10 (8): 1399–1402. doi:10.1063/1.1664979. ISSN 0022-2488.