Multicritical point with infinite fractal symmetries

Abstract

Recently, ``Pascal's triangle model'' constructed with $\mathrm{U}(1)$ rotor degrees of freedom was introduced and it was shown that ($\mathit{i}$) this model possesses an infinite series of fractal symmetries and ($\mathit{ii}$) it is the parent model of a series of ${Z}_{p}$ fractal models each with its own distinct fractal symmetry. In this work we discuss a multicritical point of Pascal's triangle model that is analogous to the Rokhsar-Kivelson point of the better known quantum dimer model. We demonstrate that the expectation value of the characteristic operator of each fractal symmetry at this multicritical point decays as a power law of space and this multicritical point is shared by the family of descendent ${Z}_{p}$ fractal models. Afterwards, we generalize our discussion to a $(3+1)d$ model termed ``Pascal's tetrahedron model'' that has both planar and fractal subsystem symmetries. We also establish a connection between Pascal's tetrahedron model and the $\mathrm{U}(1)$ Haah's code.