Thermal melting in 2D classical and quantum systems: a tensor network study

We investigate and characterise phase transitions driven by thermal fluctuations out of ordered phases into disordered ones on a series of two-dimensional models. We rely on tensor networks algorithms that approximate directly the partition function in the thermodynamic limit, and circumvent finite-size effects from which more conventional approaches can suffer.

We first revisit the problem of commensurate-incommensurate transitions in the context of two-dimensional statistical physics. In 1982, Huse and Fisher first suggested the existence of a direct chiral transition that characterises the melting of $p \times 1$ commensurate phases into incommensurate ones for $p=3$ and $p=4$. Despite the prediction being over forty years old, definitive evidence for the existence of the transition is still lacking. We investigate the issue with the corner transfer matrix renormalisation group (CTMRG) algorithm for the $p=3$ and $p=4$ cases by studying respectively the three-state chiral Potts model and the chiral Ashkin-Teller model. We argue for the existence of a chiral melting in both models. For $p=3$, we predict the chiral transition to be characterised with critical exponents $\nu_y = 1, \nu_x = 2/3, \bar \beta = 2/3$ and $\alpha = 1/3$, leading to the anisotropy exponent $z = 3/2$.

Next, we investigate the thermal properties of the Shastry-Sutherland model. Since the discovery of its experimental realisation in the SrCu$_2$(BO$_3$)$_2$ compound, the model has attracted considerable attention and its ground state phase diagram is reasonably well understood. Yet, its thermal properties remain largely unexplored. We investigate the nature of the melting of the 1/3 plateau in the Shastry-Sutherland model with infinite projected entangled-pair states (iPEPS). The ground state in the plateau breaks the $\mathbb{Z}_2$ rotational and $\mathbb{Z}_3$ translational symmetries. Two scenarios are possible: either the ordered phase melts in a two step transition where the symmetries are restored at different temperatures, or it melts in a single transition, where both symmetries are restored at a unique temperature. We provide strong numerical evidences in favour of the second scenario and we argue for the melting to occur via a single weakly first-order transition. Focussing on the experimentally relevant coupling ratio $J'/J=0.63$, our simulations predict the transition in the middle of the 1/3 plateau in the SrCu$_2$(BO$_3$)$_2$ compound to occur at $T = 4.8$K.

Finally, we introduce a variant of the CTMRG algorithm to contract tensor networks on the honeycomb lattice. The algorithm is benchmarked on the challenging antiferromagnetic triangular Ising model in a field. At finite field, we map the transition line out of the ordered phase. As a further benchmark, the algorithm is used in the context of a two-dimensional quantum many-problem. The low temperature properties of the anisotropic Heisenberg model on the maple leaf lattice are investigated with iPEPS.