Staggered operator withtopological SU(2) backgrounds atnonzero chemical potential

R. Rodlin collaboration withF. Bruckmann, T. Sulejmanpasic

University of Regensburg

02.08.2013

R. Rodl — University of Regensburg LATTICE 2013

OutlineContinuum - Instantons and zero modes at µ, T > 0.Test of staggered Dirac operator at µ, T > 0.

More information in “Topological zero modes at nonzerochemical potential”, arXiv:1305.1241 (PRD in press)

The KvBLL caloron with SU(2) gauge groupCaloron: (anti) self dual solution of YM-Theory T > 0.Explicit construction: KvBLL (Kraan, van Baal, Lee and Luhep-th/9805168, hep-th/9802108).

Important properties:

Action density is equal to its topological density (self dual).

Two monopole constituents.

Novel holonomy parameter ω.

Figure: Action density of a SU(2) KvBLL caloron (ω = 1/4, ρ = 1β)(Taken from hep-th/9805168).

R. Rodl — University of Regensburg LATTICE 2013

Continuum theory - SetupDirac operator in the fund. rep. with µ

/D(µ) = γνDν − γ0µ , Dν = ∂ν + iAν .

Eigenproblem: /D(µ)ψn(x;µ) = λn(µ)ψn(x;µ)Density and an inner product (biorthonormalization):

ρmn(x;µ) ≡ ψ†m(x;−µ)ψn(x;µ),

∫d4x ρmn(x;µ) = δmn.

What happens if Aν is a charge one SU(2) caloron?(F. Bruckmann, R. Rodl, T. Sulejmanpasic, ’13)

Density ρmn(x;µ) is real (SU(2) property).

Zero eigenvalue λ0(µ) = 0 exists, for instantons see1.

ρ00(x;µ) has negative regions at µ > 0.

The localization of the zero mode is still on top of a caloron lump.1(C. Aragao de Carvalho ’81, A. Abrikosov ’83, M. Cristoforetti ’11)

R. Rodl — University of Regensburg LATTICE 2013

Continuum theory - ResultsZero mode density (Zmd) in a background of a caloron (µ = 4Tand ω = 1/4) together with the caloron action density.

Figure: Colored magnitude is rescaled.

Zmd localization is not accidentally on top of one lump.(M. Garcia-Perez et al. ’99)

Application of the Index-Theorem1 at µ and T .

1(T. Kanazawa, T. Wettig, N. Yamamoto ’11), (R. Gavai, S. Sharma ’10)R. Rodl — University of Regensburg LATTICE 2013

Lattice - Setup and results

Strategy of the comparison - Lattice vs. continuum

Central question:Is the staggered operator suitable to reproduce topologicalzero modes at non zero chemical potential andtemperature or not?

Investigate:Spectrum of the operator, especially the (quasi-)zeroeigenvalues.The (quasi-)zero mode density constructed from thestaggered modes.

R. Rodl — University of Regensburg LATTICE 2013

Discretization scheme

Classical lattice gauge links via gauge transporter:

Aµ (x) =1

2η3µν ∂ν

(log[Φ (x)

])τ3 + ...,

Uν(an) = P exp

i

1∫0

~Aν(an+ asν) · ~τ ads

= limN→∞

N∏k=1

exp

(iaN−1 ~Aν(an+ a

k

Nν) · ~τ

).

Standard staggered operator:

2aD(µ) =∑ν

ην(n)(

Uν(n) δn+ν,meaµδν,4 −U†ν(n− ν) δn−ν,me

−aµδν,4).

R. Rodl — University of Regensburg LATTICE 2013

Discrete action density and boundary artifacts

Wilson action density suffers from boundary artifacts dueto the periodicity requirement of the gauge links.APE smearing reduces this cutoff artifacts αAPE=0.45.(M. Falcioni et al. ’85), (T. DeGrand, A. Hasenfratz, T. G. Kovacs ’97)

R. Rodl — University of Regensburg LATTICE 2013

Figure: Action density of SU(2) caloron before smearing.

Figure: Action density after 40 APE smearing steps.

Staggered operator with caloron background and µ = 0

D(µ = 0) is anti-hermitian↔ purely imaginary eigenvalues.4 tastes↔ 4 (quasi-)zero eigenvalues.

10−16× 10−2 7× 10−2 8× 10−2 9× 10−2aT

10−1510−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−2

Im(λ 0)/T

No smearingAPE smearing

Conclusion: Exact zero modes in the continuum limit.

Spectrum of the staggered operator with non zero µ

−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8Re(λ)/T

−3

−2

−1

0

1

2

3Im(λ)/T 4 xO(10−6) i

4 xO(10−6) i4 xO(10−6) i

µR = 0.02 · TµR = 0.40 · TµR = 0.80 · TµR = 0.80 · T (free)

Eigenvalues became complex.Quartet due to {η5,D(µ)} = 0 and SU(2) symmetry.Exited caloron spectrum similar to the free case.(Quasi-)zero eigenvalues at real chemical potential!

Zero eigenvalues at non zero µ

10−16× 10−2 7× 10−2 8× 10−2 9× 10−2aT

10−1510−1410−1310−1210−1110−1010−910−810−710−610−5

Im(λ 0)/T

µ = 0.90 · Tµ = 0.45 · T

Continuum limit of Im(λ0)/T with different µ.APE smeared caloron backgrounds with ω = 1/4.

Conclusion: Exact zero modes even with µ (continuum).

So far ...The staggered Dirac operator with APE smeared caloronbackground has four zero modes (in the continuum limit).This is true even at non zero real chemical potential.

... and nowWe can investigate zero mode densities constructed fromstaggered eigenvectors.

Staggered zero mode density with µ = 0

z/β0.0 1.0 2.0 3.0 4.0

x/β

0.01.0

2.03.0

ρ(x,1.5β,z,0.5β;µ

=0)

0.2

0.4

0.6

0.8

1.0

Compare one slice with the continuum.Zmd of the staggered operator agrees with its continuumcounterpart.

Remark: Take one of 4 possible zmd of the staggered operator.

Staggered zero mode density with µ = 2.4 · T

z/β0.0 1.0 2.0 3.0 4.0

x/β

0.01.0

2.03.0

ρ(x,1.5β,z,0.5β;µ

R)

0

1

2

3

4

5

6

7

Zmd is still localized (stronger!) to the same caloron lump.Staggered version has characteristic negative regions.

Remark: This zmd is constructed from 2 staggered densities(degenerate eigenvalues).

Comparison of zero mode densities - Lattice vs. continuum

1.5 1.0 0.5 0.0 0.5 1.0 1.5

x/β

1

0

1

2

3

4

5

6

7

8

ρ00/T

4

Holonomy ω=0.250, µ=2.4 ·TContinuum

Staggered

1.5 1.0 0.5 0.0 0.5 1.0 1.5

x/β

2

0

2

4

6

8

10

12

ρ00/T

4

Holonomy ω=0.125, µ=2.4 ·TContinuum

Staggered

1.5 1.0 0.5 0.0 0.5 1.0 1.5

x/β

5

0

5

10

15

20

ρ00/T

4

Holonomy ω=0, µ=2.4 ·TContinuum

Staggered

Good agreement, small discrepancy due to cutoff effects.Larger µ↔ numerically hard.

Conclusion and outlook

ConclusionThe staggered operator is suitable to reproduce topologicalzero modes at non zero chemical potential andtemperature.

OutlookExtension to SU(3) gauge group (complex density).Continuum expressions of the zero mode density for thesemiclassical approach to QCD at T, µ > 0 (overlap matrixelements).

R. Rodl — University of Regensburg LATTICE 2013

We useARPACKBoostArmadillo

Thank you!