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The multistate Potts models on two-dimensional hyperbolic lattices arestudied with respect to various boundary effects. The free energy isnumerically calculated by Corner Transfer Matrix Renormalization Group method.We analyze phase transitions of the Potts models in the thermodynamic limitwith respect to contracted boundary layers. A false phase transition is presenteven if a couple of the boundary layers are contracted. Its significanceweakens, as the number of the contracted boundary layers increases, until thecorrect phase transition (deep inside the bulk) prevails over the false one.For this purpose we derive a thermodynamic quantity, the so-called bulk excessfree energy, which depends on the contracted boundary layers and memorizesadditional boundary effects. In particular, the magnetic field is imposed onthe outermost boundary layer. While the boundary magnetic field does not affectthe second-order phase transition in the bulk if suppressing all the boundaryeffects on the hyperbolic lattices, the first-order (discontinuous) phasetransition is significantly sensitive to the boundary magnetic field. Contraryto the phase transition on the Euclidean lattices, the discontinuous phasetransition on the hyperbolic lattices can be continuously controlled (within acertain temperature coexistence region) by varying the boundary magnetic field.

Boundary-field-driven control of discontinuous phase transitions on hyperbolic lattices