Yuji Hirono
Topological order and color superconductivity
Asia Pacific Center for Theoretical Physics
In collaboration with Yuya Tanizaki [1811.10608]
Phases of QCD matter
T
baryon chemical potential Neutron stars
Early universe
Heavy ion collisions Quark-gluon plasma
Color superconductivity
Hadrons
“Quark-hadron continuity”
u d s
Nucleon superfluidity
Color superconductor
?
“CFL phase”
[Schafer, Wilczek '99]
“Topological order”
Outline
• Color SC & quark-hadron continuity
• Vortices in CFL phase
• Low-energy effective gauge theory
• BF theory + massless phonons
!4
Color superconductivity
• Order parameter: diquark condensate
• Symmetry transformation
• Three light quarks with degenerate mass
• up, down, strange
color flavor
Color-flavor locked phase
!6
• Ground state
u d s
• All the gluons are gapped: color SC
• SSB of global U(1): superfluidity
“Quark-hadron continuity”
• Global symmetry
U(1) phonons
Pions
• Light modes - Nambu-Goldstone bosons The same in nucleon superfluid & CFL
[Schafer, Wilczek '99]
“Quark-hadron continuity”
!8
Pions
Vector mesons
CFL pions
Gluons
Baryons Quarks
U(1) phonons U(1) phonons
[Yamamoto, Tachibana, Hatsuda, Baym '07]
[Fukushima, Hatsuda '10]
……
Topological order
• Order that cannot be captured by local order parameters
• Robust degeneracy of ground states
• Ex) s-wave SC / FQHE
• Phase transition is needed btw. states with different topological order
• Order parameter: Wilson loop, etc
• “Higher-form symmetry”
and its spontaneous breaking
[Gaiotto, Kapustin, Seiberg, Willett ’15]
[X. G. Wen ’90]
Vortices in CFL
!10
• Quantized (1/3) superfluid circulation
• Color magnetic flux
[http://cua.mit.edu/ketterle_group/Nice_pics.htm]
• Rotating neutron star
[Arata Yamamoto’s talk tomorrow]
[Balachandran, Digal, Matsuura '06]
Fractional statistics of vortices & particles
[Cherman, Sen, Yaffe 1808.04827]
Z3 braiding phases
=
color Wilson loop
u d s
Nucleon superfluidity
Color superconductor
[Cherman, Sen, Yaffe 1808.04827]
!12
Z3 braiding phases
“CFL phase”
Fractional statistics of vortices & particles
Low-effective theory for CFL
• We consider degenerate masses for u, d, s
• Massless degrees of freedom:
U(1) phonons
• Correlation of U(1) circulation & color holonomy
• Fractional statistics
Dual effective gauge theory - SC
!14
BF theory for superconductivity
• Abelian Higgs model
• EOM for :
BF theory for superconductivity
!16
• Physical observables
Wilson loop operator Vortex operator
: world line of a quasiparticle : world-sheet of a vortex
BF theory for superconductivity
Fractional statistics of quasiparticles & vortices
=
• Emergent Z_k 1-form & 2-form symmetry
BF theory for superconductivity
!18
BF theory for superconductivity
• Emergent Z_k 1-form & 2-form symmetry
BF theory for superconductivity
!20
• 1-form & 2-form symmetries are
spontaneously broken
“Topological order”
BF theory for superconductivity
Z_k Fractional Braiding phase
1-form symmetry
2-form
SSB
SSB
Topological Order
Dual effective
gauge theory - CFL
!22
[Hirono, Tanizaki 1811.10608]
GL model for CFL
• GL Lagrangian
• Drop amplitude fluctuations kinetic term of the gauge field
• Fix the gauge so that
Dual theory for CFL
!24
• Topological BF theory coupled with massless superfluid phonons
• K matrix
• not square
• dim coker K = (# of massless phonons)
Phonons BF term
Dual theory for CFL
• Physical observables
Phonons
Dual theory for CFL
!26
is the Moore-Penrose inverse of Massless phonons
Dual theory for CFL
Physical charge vectors
Dual theory for CFL
!28
Discrete (Z3) 2-form symmetry
No discrete 1-form symmetry
2-form symmetry is unbroken
No topological degeneracy of the ground states
• Z3 2-form symmetry 2-form symmetry
• Continuous 2-form symmetry cannot be broken in 4D (Coleman-Mermin-Wagner theorem)
• Vortices are log-confined because of massless phonons
• p-form symmetry cannot be broken if d - p ≦ 2
Summary
!30
Z3 Fractional Braiding phase
1-form symmetry
2-form
symmetry
SSB
massless phonons
u d s
Nucleon superfluidity
Color superconductor
“CFL phase”
