Higgs-confinement transitions in QCD from symmetry protected topological phases

In gauge theories with fundamental matter there is typically no sharp way to distinguish confining and Higgs regimes, e.g. using generalized global symmetries acting on loop order parameters. It is standard lore that these two regimes are continuously connected, as has been explicitly demonstrated in certain lattice and continuum models. We point out that Higgsing and confinement sometimes lead to distinct symmetry protected topological (SPT) phases – necessarily separated by a phase transition – for ordinary global symmetries. We present explicit examples in 3+1 dimensions, obtained by adding elementary Higgs fields and Yukawa couplings to QCD while preserving parity \mathsf{P} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖯</mml:mi> </mml:mstyle> </mml:math> and time reversal \mathsf{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> </mml:math> . In a suitable scheme, the confining phases of these theories are trivial SPTs, while their Higgs phases are characterized by non-trivial \mathsf{P} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖯</mml:mi> </mml:mstyle> </mml:math> - and \mathsf{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> </mml:math> -invariant theta-angles \theta_f, \theta_g = \pi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> for flavor or gravity background gauge fields, i.e. they are topological insulators or superconductors. Finally, we consider conventional three-flavor QCD (without elementary Higgs fields) at finite U(1)_B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> </mml:math> baryon-number chemical potential \mu_B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:math> , which preserves \mathsf{P} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖯</mml:mi> </mml:mstyle> </mml:math> and \mathsf{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> </mml:math> . At very large \mu_B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:math> , three-flavor QCD is known to be a completely Higgsed color superconductor that also spontaneously breaks U(1)_B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> </mml:math> . We argue that this high-density phase is in fact a gapless SPT, with a gravitational theta-angle \theta_g = \pi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> that safely co-exists with the U(1)_B <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> </mml:math>