Dimer-Quadrupolar Quantum Phase Transition in the Quasi-1D Heisenberg model with Biquadratic Interaction

by Kenji Harada, Naoki Kawashima and Matthias Troyer

Journal club talk by Christoph Puetter
Summary below by Thomas Grzesiak



Christoph Puetter introduced the paper "Dimer-Quadrupolar Quantum Phase Transition in the Quasi-1D Heisenberg model with Biquadratic Interaction" by Harada, Kawashima, and Troyer. He explained that the concept of a deconfined quantum critical point (DQCP) is only a few years old, and people are still looking for models which exhibit this kind of phase transition. This paper investigates 2 critical points in the phase diagram of the (anisotropic) bilinear biquadratic Heisenberg model with Jl the bilinear coupling strength and Jq for the biquadratic term. For |Jl|>> |Jq|, we get the familiar ferromagnetic
or Neel state depending on the sign of Jl. When the couplings are comparable in strength, the situation becomes more complex.

To illustrate this point, Christoph began by considering 2 isolated sites and showed that a singlet is preferred for Jl < Jq < 0. He explained that once more sites are included the singlet condition can no longer be satisfied on bonds sharing the same site. The system therefore develops new phases such as the dimer phase (for &lambda << 1) and the spin nematic. (lambda is the anisotropy parameter in the 2d Heisenberg model considered here).

The authors of the paper performed MC simulations for various lattice sizes L, and measured the Quadrupole-Quadrupole (for nematic) and Dimer-Dimer correlation functions, Gq and Gd. They examined the ratio G(L/2)/G(L/4). Tuning lambda they found that these ratios (Rq and Rd) become independent of L at lambda_critical, signalling the phase transition.

It was asked why the above ratio was chosen. Daniel explained this was for the correlation length, to get a constant alpha independent of L (a power law). Christoph then focused on a special point on the nematic-dimer phase boundary (Y in the paper). He explained that with spin-rotational symmetry preserved and translational symmetry broken in the dimer phase, and vice versa in the nematic, a first order transition is expected. It is also expected that Rq and Rd should be 1 (why?), and that there is a cusp in the energy at the transition. However, the fact that this was not observed could be due to a DQCP, although other possibilities cannot be ruled out.

It was asked why DQCP doesn't appear from the Neel side (point X in the paper)? If there are 2 ordered states with no relationship between the orderings, there should be a first order transition, according to Landau, yet this paper found a second order transition. The talk ended with another question, as to what a DQCP actually is (in a nutshell). Michael gave an example: in a magnetic phase has spin waves that are spin 1 excitations in a spin 1/2 system. They can split into 2 spin 1/2 excitations at a DQCP, ie they become deconfined. For the dimer phase, spins are bound into singlet states. At a DQCP, they would become deconfined, breaking the singlet.