Spin-1/2 J1-J2 model on square lattice

September 7, 2020

Peolpe have been working on J1-J2 model for 30 years. Here is a recent tensor network calculation. Comparing with the following meanfield result may give us some hint on what is the discovered gapless spin liquid state near J1=J2.


The meanfield phase diagram of J1-J2 model from a SU(2) projective construction obtained in Quantum Orders and Symmetric Spin Liquids (J1 =1-J2). It contains different types of spin liquids with low meanfield energies.

Phase C is phase A (SU2 pi-flux) on each sublattice. Phase B is phase I (SU2 uniform) on each sublattice.

  • Phase D is the gapped chiral spin state proposed for the same J1-J2 model.
  • Phases A, C, E, G, H are gapless with linear spin-1 excitations (i.e. spin-1/2 spinons are massless Dirac fermions).
  • Phases B, I are gapless with spin-1/2 spinon Fermi surface. 
  • Phase F is spin gapped but has gapless spin-0 U(1) gauge bosons. The gapless U(1) gauge theory is unstable in 2+1D, which may become a gapped dimmer phase.


The low energy edge of spectral weight of spin-1 excitations in different spin liquid phases:

(a) SU2Bn0 (phase A), (b) U1Cx10x

(a) Z2A0013 (phase H), (b) Z2Azz13 (phase G), (c) Z2A001n, (d) Z2Azz1n

(a) Z2B0013, (b) Z2Bzz13, (c) Z2B001n, (d) Z2Bzz1n

(a) Z2Ax2(12)n, (b) Z2Bx2(12)n

(a) U1Cn01n, (b) U1Cn00x (phase F, the spinons can be gapped or gapless. The gapless case is plotted here, but the gapped case appears in meanfield phase diagram.)

(a) U1Cn0x1, (b) U1-linear in Eq. 123 (phase E)

(a) U1B0001, (b) U1B0011


Those different spin liquids are characterized by the projective symmetry groups (PSG's) of the spin-1/2 fermionic spinons. The topological excitations in a topological order sometimes can be viewed as particles carrying gauge charge of a gauge group (called invariant gauge group IGG). Under this point of view, the particle actually has a symmetry described by group PSG. Mathematically, PSG is an extension of symmetry group (SG) by the gauge group (IGG):  1 → IGG → PSG → SG → 1. In the above meanfield theory, SG include the x- and y-translation, the point group of square lattice, and spin rotation symmetries.


In general, a topological order with symmetry is characterized by several PSG's, one for each type of topological excitation.