Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

100 fractional electric charge and quark confinement
Ergodicity becomes a tricky business, owing to the additional topology that has
been introduced via the U(1). HMC trajectories only propose small changes to the
link variables and have difficulty changing the sign of the U(1) links,
e iα µ
→ −e iα µ
∈ U(1) (5.31)
For large β em , the U(1) gauge action strongly suppresses fluctuations away from
links that are gauge equivalent to ±1 ∈ U(1), i.e. e i2α µ
= 1. Additional local updates
are necessary to ensure ergodicity.
The first of these is a simultaneous flip of SU(2) and U(1) links,
U µ (x) → −U µ (x); e iα µ(x) → −e iα µ(x) . (5.32)
Only the SU(2) Wilson gauge action is affected by this combined update. The U(1)
gauge action is blind to such phases by default, and the quark determinant is
also blind because this transformation belongs to the Z 2 symmetry of the SU(2) ×
U(1) model. It introduces a thin combined color-electromagnetic vortex that pierces
the plaquettes belonging to the affected link, which allows for tunneling between
topological sectors. The color-flux is subsequently free to spread. Between HMC
trajectories we propose such an update on a random link and accept/reject using a
Metropolis check [149] for the SU(2) gauge action, repeating many times.
This is efficient when β col is small, but the acceptance rate is exponentially suppressed
by the Boltzmann weight e −S as the cost of flipping SU(2) plaquettes increases
with β col . Thin color vortices become expensive. It is necessary to also include
local updates that flip only U(1) links,
e iα µ(x) → −e iα µ(x) . (5.33)
Here the fermion action requires a costly recalculation for each Metropolis check,
but the acceptance rate is much improved when β col is large. In what follows, we
have tuned the the number of sweeps of each type of local update to obtain an
acceptable compromise for different parameter regions, and carried out simulations
from both ordered starts (i.e., all links are unity), and disordered starts (i.e., all links
are random) for confirmation of ergodicity.
5.3.2 Dynamical restoration of symmetry
So what are the consequences of coupling quarks with fractional charge to a U(1)
gauge group in this model? The usual ordering of color links by dynamical quarks
is negated, which is best understood via the hopping expansion. To leading order
in the hopping parameter with N t = 4 time slices, our Wilson quarks contribute
the O(κ 4 ) effective action,
( )
S f ,eff = −16κ 4 ∑ cos α µν · tr □ col + 8 ∑ ReP em · P col + . . . , (5.34)
plaq.
⃗x
These minimal quark loops generate plaquette-plaquette and Polyakov loop-Polyakov
loop couplings between the gauge groups. The trivial P col = 1 sector is no longer
favored, provided that there is disorder in the U(1) Polyakov loop P em . From the