Critical behavior of 3D <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> lattice gauge theories at zero temperature

Three-dimensional <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> lattice gauge theories at zero temperature are studied for various values of N . Using a modified phenomenological renormalization group, we explore the critical behavior of the generalized <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> model for <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>8</mn></math> . Numerical computations are used to simulate vector models for <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>20</mn></math> for lattices with linear extension up to <math altimg="si4.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mn>96</mn></math> . We locate the critical points of phase transitions and establish their scaling with N . The values of the critical indices indicate that the models with <math altimg="si5.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>&gt;</mo><mn>4</mn></math> belong to the universality class of the three-dimensional XY model. However, the exponent α derived from the heat capacity is consistent with the Ising universality class. We discuss a possible resolution of this puzzle