We have learned instantons, monopoles and vortices in previous sections. In this section, we will introduce the methods to find monopoles and vortices on the lattice by Abelian and center projections respectively. Instantons can be found on the lattice after cooling/smearing etc. [43].
2.9.1 Direct maximal center gauge
We know in section 2.3 that vortices are quantised magnetic fluxes. In a gauge groupG, the quantized magnetic fluxes ΦB of
eigΦB=ZG, (2.82)
whereZG are center elements ofG, are called center vortices.
Figure 2.8: An SU(2) example of a center configuration and its vortices. The dashed lines on the plane are trivial +1 links, while the solid lines on the plane are −1 links, and the fat line is the vortex penetrating this plane. In short, vortices penetrate the non-trivial plaquettes (inSU(2) just−1 plaquettes).
To find the center vortices on lattice, one needs to project the gauge config-urations onto center degrees of freedom by removing the non-confining fluctua-tions which are supposed to have no influences on the far-infrared physics [6].
Direct maximal center gauge (DMCG) [44] is an intuitive method to reveal the center vortices. It gauge transforms the configuration of adjoint links as close to a trivial configuration (all the links are1) as possible, because center elements of a groupGare the identity in the adjoint representation.
We maximize
R=X
x,µ
Tr[UAµ(x)], (2.83)
whereUAµ are links in the adjoint representation. InSU(N), the expression of R can be simplified to
R=X
x,µ
Tr[Uµ(x)]Tr[Uµ†(x)] +const. (2.84)
by virtue ofP
k(Tk)ab(Tk)cd= 12δadδbc−2N1 δabδcdand (UA)ab= Tr[TaU TbU†].
After the gauge transformation by maximizing R, the configuration (in fun-damental representation) is in the possible closest form to a center configuration.
We decompose the links into the center part Zµ and the remaining fluctuation partVµ as
Uµ(x)→Zµ(x)Vµ(x). (2.85) In SU(2), this decomposition is done by Zµ(x) = sign(Tr[Uµ(x)]). Note that the vortices are not the non-trivial center element links, but the plaquettes on the dual lattice that penetrate the non-trivial plaquettes as shown in Fig. 2.8.
An advantage of DMCG is that if one “inserts” some center vortices to the configuration by hand, which means to change some links U → ZkU, these
’inserted’ vortices will be detected because the configuration in the adjoint rep-resentation is the same to that before the insertion.
DMCG has a weakness, the Gribov copy problem [45]. It happens in the procedure of maximizing R in eqn. (2.84), that R as a function of the gauge transformation has many local maxima. In numerical simulations, it is practi-cally impossible to find the global maximum and one ends up in a local one.
This influences the location and further properties of center vortices.
2.9. FINDING TOPOLOGICAL OBJECTS ON THE LATTICE 27
2.9.2 Indirect maximal center gauge
Indirect maximal center gauge (IMCG) [46] is very useful in exploring the con-nection between Abelian monopoles and vortices [6]. It has two steps, the first step is the maximal Abelian gauge (MAG) [47], which minimizes the non-diagonal elements of the links. InSU(2), it maximizes
R=X
x,µ
Tr[Uµ(x)σ3Uµ†(x)σ3], (2.86) this is equivalent to minimizing the non-diagonal elements because
1
2Tr[σµσ3¯σνσ3] =
1 (µ=ν= 0,3) 0 (µ6=ν)
−1 (µ=ν= 1,2)
. (2.87)
After having maximizedR, one decomposes the linksUµ0(x) =Cµ(x)Dµ(x) into an Abelian partDµ(x) and the background partCµ(x) where
Dµ= 1
q|Uµ110 ||Uµ220 |
Uµ110 0 0 Uµ220
. (2.88)
Exploiting the remaining U(1) gauge freedom, which amounts to a shift θµ(x)→αθµ(x) =−α(x) +θµ(x) +α(x+ ˆµ), whereθµ(x) = arg (Uµ(x)11), the second step maximizes the IMCG functional
FIMCG[U] =X
µ,x
(cos(αθµ(x)))2 , (2.89) that serves the same purpose as theRin eqn. (2.83).
Finally, the projectedZ(2) gauge links are defined as
Zµ(x) = sign (cos(αθµ(x))) . (2.90)
2.9.3 Laplacian center gauge
Laplacian center gauge (LCG) [41], which avoids the Gribov copies problem, utilizes the two eigenmodesφ(1)andφ(2)of the gauge covariant lattice Laplacian operator in the adjoint representation:
−∆[UA]φ(1,2) = λ1,2φ(1,2) (2.91)
∆abxy[UA] = 1 a2
X
µ
UAµ(x)abδx+ ˆµ,y+UAµ(x−µ)ˆ baδx−ˆµ,y
−2δabδxy
a, b= 1,2, nA, (2.92)
where nA is the dimension of adjoint representation. Consider that vortices should reflect the infrared feature of the gauge field, the lowest two eigenmodes are preferred.
The first step of LCG, which is also called Laplacian Abelian gauge (LAG) [48], transformsthe lowest eigenmodeΦ(1)=φ(1)a Ta diagonal,
Φ(1)0(x) = Ω(x)Φ(1)(x)Ω†(x) Φ(1)i,j0(x) = 0 (i6=j). (2.93)
The gauge transformation Ω(x) is not unique, it is easy to find that gauge trans-formations keeping Φ(1)0 diagonal, namely in the Cartan subgroup of SU(N) matrices, are
V(x) = exp
diag iα1(x), ..., iαN(x) ,
N
X
i=1
αi(x) = 0. (2.94) An additional freedom we need to fix in the first step is the exchange of the non-diagonal entries of Φ(1)0 – the eigenvalues of Φ(1). The gauge transformations exchanging the eigenvalues are not in the U(1)N−1 Cartan subgroup, so we should fix the order of the eigenvalues ascending or descending.
We can also identify the remaining symmetry after the first step of LCG by the number of free parameters. AnSU(N) matrix hasN2−1 real parameters, from eqn. (2.93) we can find the number of real conditions isN(N−1)/2×2 = N(N−1). So the remaining number of free parameters is N−1 and Cartan subgroup is the remaining symmetry.
As the first step has restricted the remaining symmetry to the Cartan sub-group, the second step fixes the remainingU(1)N−1 symmetry toZN. Now we see why to choose the adjoint representation, because the adjoint fields are blind to the centers ofSU(N)
ZkΦ(1,2)Zk†= Φ(i) (2.95)
The first excited eigenmode after the first step is
Φ(2)0(x) = Ω(x)Φ(2)(x)Ω†(x). (2.96) The second step applies the gauge transformationV(x)∈U(1)N−1to Φ(2)0(x), Φ(2)00(x) =V(x)Φ(2)0(x)V†(x). (2.97) The remainingU(1)N−1 symmetry is fixed byrotatingN−1 non-diagonal en-tries of Φ(2)00 real and positive.
Defects can occur in the procedure of LCG. In the first step, if Φ(1) has two equal eigenvalues, or in the second step, if any of theN−1 non-diagonal entries of Φ(2)0 vanishes.
The defects of these two LCG steps can be related to monopoles and vortices.
The defects of the first step is very obvious in anSU(2) monopole configuration, whose lowest eigenmode is a zero mode, the scalar fieldφitself (see section 3.3.1) which vanishes at the center of the monopole. So we expect the defects in the first LCG step to be the world lines of monopoles. Indeed, the hedgehog like lowest mode has an S2 winding number, the first LCG step rotates this lowest mode to a fixed direction meaning the gauge transformation Ω around the monopole reflects an S2 winding, which will appear in the Aµ from this gauge transformation. InSU(N), the defect promotes the remaining symmetry fromU(1)N−1 toSU(2)×U(1)N−3.
Let us consider the defects of the second LCG step in SU(2). If one finds Φ(2)1,20(xV) = 0 on the plane that passes through xV and extends in the e1 = Oφ(2)1 0(xV) and e2 = Oφ(2)2 0(xV) directions (O is the gradient operator), the Taylor expansion of Φ(2)1,20 aroundxV is
Φ(2)1,20(x) = (x−xV)·e1+i(x−xV)·e2+O((x−xV)2)≈f(r, θ)eiθ, (2.98)
2.10. TOPOLOGICAL OBJECTS AND THE QCD PHASE TRANSITION29