Arbitrary number of thermally induced phase transitions in different universality classes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:math> models with higher-order terms

We propose generalized variants of the XY model capable of exhibiting an arbitrary number of phase transitions only by varying temperature. They are constructed by supplementing the magnetic coupling with n_{t}-1 nematic terms of exponentially increasing order with the base q=2,3,4, and 5, and increasing interaction strength. It is found that for q=2,3, and 4 with sufficiently large coupling strength of the final term, the models exhibit a number of phase transitions equal to the number of the terms in the generalized Hamiltonian. Starting from the paramagnetic phase, the system transitions through the cascade of n_{t}-1 nematic phases of the orders q^{k}, k=n_{t}-1,n_{t}-2,⋯,1, that are characterized by q^{k} preferential spin directions symmetrically disposed around the circle, to the ferromagnetic (FM) phase at the lowest temperatures. Besides the Berezinskii-Kosterlitz-Thouless (BKT) transition from the paramagnetic phase, all the remaining transitions have a non-BKT nature: depending on the value of q they belong to either the Ising (q=2 and 4) or the three-state Potts (q=3) universality class. For q=5, due to the interplay between different terms, the phase transitions between the ordered phases observed for q<5 split into two, and the number of the ordered phases increases to 2n_{t}-1. These phases are characterized by a domain structure with the gradually increasing short-range FM ordering within domains that extends to different kinds of FM ordering in the last two low-temperature phases. The respective transitions do not seem to obey any universality.