The spin-1/2 frustrated J1-J2 Heisenberg model on the two-dimensional (2D) square lattice (where J1 and J2 are the strengths of the first and second neighbor couplings Si - Sj , respectively) has been studied and debated since the early days of the high-Tc cuprate superconductors. The initial interest in the system stemmed from the proposal that frustrated antiferromagnetic (AFM) couplings could lead to a spin liquid (SL) in which preformed pairs (resonating valence bonds) become superconducting upon doping. Later, with frustrated quantum magnets emerging in their own right as an active research field, the J1-J2 model became a prototypical 2D system for theoretical and computational studies of quantum phase transitions and nonmagnetic states. Of primary interest is the transition from the long-range Neel AFM ground state at small g = J2/J1 to a nonmagnetic state in a window around g=0.5 (before a stripe AFM phase at g>0.6). The nature of this quantum phase transition has remained enigmatic. The nonmagnetic state may be one with spontaneously broken lattice symmetries due to formation of a pattern of singlets (a valence-bond-solid, VBS) or a SL. The quantum phase transition out of the AFM state may possibly be an unconventional “deconfined” transition. However, because of the small system sizes accessible, previous calculations was not possible to rule out a direct AFM–VBS transitions.

L. Wang from CSRC and her collaborator A. Sandvik from Boston University demonstrated an intervening gapless SL by locating the AFM–SL and SL-VBS transitions using a numerical level-spectroscopy approach, where finite-size transition points are defined using excited-level crossings. These crossing points exhibit smooth size dependence and can be more reliably extrapolated to infinite size than the order parameters and gaps used in past studies. They use a variant of the DMRG method to calculate the ground state energy as well as several of the lowest singlet, triplet and quintuplet excited energies. They demonstrated a singlet-triplet level crossing in the J1-J2 model which for 2L by L cylindrical lattices shifts as gc2 - gc2 (L) ∝1/ (L*L) and converges to gc2=0.52. In addition they observed a singlet-quintuplet level crossing, which converges to a different point, gc1=0.46. Given an analogy in a Heisenberg spin chain with long rang interactions, where singlet-quintuplet and singlet-triplet level crossings are associated with AFM-SL and SL-VBS transitions respectively, they interpret both gc1 and gc2 as quantum-critical points. For gc1 < g < gc2 the system appears to be a gapless SL with algebraically decaying correlations.

Their work demonstrated a new powerful generic way to use the DMRG method, with the demonstration itself solving a long-standing important problem in quantum magnetism and presenting new spectral properties that should be useful for understanding SLs beyond the one studied there.**Fig.1: **Gaps to the relevant S = 0, 1, and 2 excitations vs g for L = 10. The insets show the regions of the level crossings of interest for L = 6,8,10 (gaps decreasing with increasing L). The curves show polynomial fits.

**References:**

[1] Ling Wang and Anders Sandvik, Phys. Rev. Lett. 121, 107202 (2018)