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.

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

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

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

• Néel phase (${\displaystyle 0\lesssim J_{2}/J_{1}\lesssim 0.4}$): Long-range antiferromagnetic order similar to the Heisenberg model, though with reduced staggered magnetization due to quantum fluctuations.²
• Quantum paramagnetic / spin liquid phase (${\displaystyle 0.4\lesssim J_{2}/J_{1}\lesssim 0.6}$): 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.³
• Collinear striped phase (${\displaystyle J_{2}/J_{1}\gtrsim 0.6}$): Magnetic order returns, but with the collinear stripe pattern.⁴

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.²