Necessary Condition for Emergent Symmetry from the Conformal Bootstrap

Area of Science:

• Theoretical Physics
• Condensed Matter Physics
• High Energy Physics

Background:

• Symmetry enhancement under renormalization group flow is crucial for understanding phase transitions.
• Discrete symmetries can potentially enhance to continuous symmetries, a phenomenon not fully understood.
• The conformal bootstrap program offers a powerful, non-perturbative method to study quantum field theories.

Purpose of the Study:

• To derive necessary conditions for emergent symmetry enhancement from discrete (Z_n) to continuous (U(1)) symmetries in three dimensions.
• To establish quantitative predictions for the scaling dimensions of order parameter fields at conformal fixed points.
• To investigate the stability of proposed fixed points and their implications for second-order phase transitions.

Main Methods:

• Application of the conformal bootstrap program to constrain theoretical models.
• Simultaneous analysis of multiple four-point functions to derive robust conditions.
• Calculation of critical scaling dimensions for various discrete symmetries (Z_2, Z_3, Z_4).

Main Results:

• For Z_2 to U(1) enhancement, the order parameter scaling dimension must satisfy Δ_1 > 1.08.
• Necessary conditions for Z_3 (Δ_1 > 0.580) and Z_4 (Δ_1 > 0.504) symmetry enhancement were obtained.
• Demonstrated that some previously proposed fixed points are unstable under perturbations allowed by discrete symmetries.

Conclusions:

• The derived conditions impose significant constraints on critical phenomena, including QCD phase transitions and critical O(n) models.
• Emergent continuous symmetry with a second-order phase transition is impossible if these necessary conditions are violated.
• The conformal bootstrap provides a rigorous framework for understanding symmetry breaking and enhancement in quantum field theories.