Many frustrated spin models on three-dimensional (3D) lattices are currently being investigated, both experimentally and theoretically, and develop new types of long-range orders in their respective phase diagrams. They present finite-temperature phase transitions, most likely in the Heisenberg 3D universality class. However, the combination between the 3D character and frustration makes them hard to study. We present here several methods derived from high-temperature series expansions (HTSEs), which give exact coefficients directly in the thermodynamic limit up to a certain order; for several 3D lattices, supplementary orders than in previous literature are reported for the HTSEs. We introduce an interpolation method able to describe thermodynamic quantities at
T>Tc, which we use here to reconstruct the magnetic susceptibility and the specific heat and to extract universal and nonuniversal quantities (for example, critical exponents, temperature, energy, entropy, and other parameters related to the phase transition). While the susceptibility associated with the order parameter is not usually known for more exotic long-range orders, the specific heat is indicative of a phase transition for any kind of symmetry breaking. We present examples of applications on ferromagnetic and antiferromagnetic models on various 3D lattices and benchmark our results whenever possible.
Continuous phase transitions exhibit richer critical phenomena on the surface than in the bulk, because distinct surface universality classes can be realized at the same bulk critical point by tuning the surface interactions. The exploration of surface critical behavior provides a window looking into higher-dimensional boundary conformal field theories. In this work, we study the surface critical behavior of a two-dimensional (2D) quantum critical Heisenberg model by tuning the surface coupling strength, and discover a direct special transition on the surface from the ordinary phase into an extraordinary phase. The extraordinary phase has a long-range antiferromagnetic order on the surface, in sharp contrast to the logarithmic decaying spin correlations in the 3D classical O(3) model. The special transition point has a new set of critical exponents, y_{s}=0.86(4) y s = 0.86 ( 4 ) and \eta_{\parallel}=-0.33(1) η ∥ = − 0.33 ( 1 ) , which are distinct from the special transition of the classical O(3) model and indicate a new surface universality class of the 3D O(3) Wilson-Fisher theory.
We study the triangular-lattice bilayer Heisenberg model with antiferromagnetic interplane coupling ${J}_{\ensuremath{\perp}}$ and nearest-neighbor intraplane coupling $J\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\lambda}{J}_{\ensuremath{\perp}}$ by expansions in $\ensuremath{\lambda}$. For negative $\ensuremath{\lambda}$ a phase transition is found to an ordered phase at a ${\ensuremath{\lambda}}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.2636\ifmmode\pm\else\textpm\fi{}0.0001$, which is in the 3D classical Heisenberg universality class. For $\ensuremath{\lambda}>0$, we find a transition at a rather large ${\ensuremath{\lambda}}_{c}\ensuremath{\approx}1.2$. The universality class of the transition is consistent with that of Kawamura's 3D antiferromagnetic stacked triangular lattice. The spectral weight for the triplet excitations, at the ordering wave vector, remains finite at the transition, suggesting that a phase with free spinons does not exist in this model.