Vortices and chiral symmetry breaking
What do we know about the mechanism of chiral symmetry breaking in SU(2) lattice QCD?
in cooperation with Urs Heller, Roman Höllwieser, Thomas Schweigler J. Greensite, and Š. Olejník
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 1 / 29 Outline
◮ Reminder of lattice QCD ◮ Notions of confinement ◮ Center vortex dominance ◮ Chiral symmetry breaking and Dirac spectrum ◮ Center vortices and topological charge ◮ Center vortices and Dirac eigenmodes ◮ Dirac spectra for spherical vortices and instantons ◮ SummaryClassical
R. Höllwieser et al., PRD78, 054508 (2008) [arXiv:0805.1846] R. Höllwieser et al., JHEP 1106, 052 (2011) [arXiv:1103.2669] T. Schweigler et al., PRD87, 054504 (2013) [arXiv:1212.3737] R. Höllwieser et al., PRD88, 114505 (2013) [arXiv:1304.1277]
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 2 / 29 SU(2) Gauge theory on a lattice Continuum Lattice
U
x x x+µ^
i ¯a τa a τa gaAµ(x) 2 Aµ(x) 2 ∈ su(2), Uµ(x) = e ∈ SU(2) 1 1 4 4 2 S = 4 d x(Fµν,a) , S = β (1 − Re Tr U ), β = 2 2 g R X
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 3 / 29 Field distributions for point charges
QED Maxwell equations QCD} perturbative vacuum}
Coulomb field
unconfined charges } Maxwell equations Non-perturbative vacuum
What is condensed in the vacuum ? Magnetic flux
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 4 / 29 Vortices and chiral symmetry breaking
◮ Vortices are quantised magnetic fluxes ◮ Vortices explain confinement ◮ Is there a relation to chiral symmetry breaking? ◮ Nambu (1960): explained low pion mass pion is a collective excitation of broken chiral symmetry ◮ Lagrangian of massless QCD in continuum: chiral symmetric ◮ hψψ¯ iMS (2 GeV)= −(250 ± MeV)3
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 5 / 29 Notions of confinement
◮ Confinement means that the spectrum consists only of color singlet states. ◮ Confinement means that no color-nonsinglet asymptotic states exist. ◮ In some cases, for theories with a center symmetry, confinement means that Wilson loops have an area law; the center symmetry is unbroken.
For theories with a center symmetry we have a true order parameter for confinement, the expectation value of a Polyakov loop, hLi,
Nt L(~x)= U4(~x, t) , world-line of static quark t Y=1
Under center transformations: L(~x) → zL(~x), with z ∈ ZN.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 6 / 29 Wilson loops R * a
T * a
W (R, T )= hTr U i a l lY∈C ◮ Wilson loop ⇐⇒ Expectation value of a static quark antiquark pair W (R, T ) = exp (−V (R)T − α(R + T )) = exp (−σRT − α(R + T )) ◮ String tension from area law 1 1 σ = lim V (R)= − lim ln W (R, T ) R→∞ R R,T →∞ RT Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 7 / 29 Area law for center projected Wilson loops
−1 −1 −1 −1 P-vortex −1 −1 −1 −1
−1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
denote f the probability that a plaquette has the value -1 hW (A)i = [(−1)f + (+1)(1 − f )]A = exp[ln(1 − 2f ) A]= exp[−σ R × T ], −σ A | {z }σ ≡− ln(1 − 2|f ){z≈ }2f Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 8 / 29 Center projection
A method to detect vortices. Fix to direct maximal center gauge, i.e., to Landau gauge in the adjoint repr. Then, center projection for SU(2) consists in
Uµ(x) → U˜µ(x)= Zµ(x)= signTr Uµ(x) , and vortex removal ′ Uµ(x)= Zµ(x)Uµ(x) . P-vortices occupy world sheets that are dual to negative plaquettes in the center-projected gauge fields. Center dominance means that long distance (nonperturbative) “physics” is reproduced by the center-projected fields. The vortex-removed gauge fields retain short-distance (perturbative) physics only.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 9 / 29 Center vortex dominance
0.35 full theory center projected 0.3 vortex removed asymptotic string tension 0.25
0.2
0.15 x(R,R) 0.1
0.05
0
-0.05 0 1 2 3 4 5 6 R
Creutz ratios for full, center-projected, and vortex-removed gauge fields for βLW = 3.3.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 10 / 29 Center vortex dominance
1 two loop L=84 L=124 L=164 L=204
P-vortex 0.1 surface density vs. βLW vortex density
0.01 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 β LW
2 “Two-loop” line is scaling prediction with ρv /6Λ = 50. Scaling shows the vortex density is a physicalq quantity, with a well defined continuum limit.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 11 / 29 Chiral symmetry breaking and Dirac spectrum
A strong enough attractive force is expected to induce spontaneous chiral symmetry breaking. A force strong enough to confine is certainly expected to do so. According to Banks-Casher, chiral symmetry breaking is necessarily associated with a finite density of near-zero modes of the chiral-invariant Dirac operator hψψ¯ i = πρ(0+) , where ρ(λ) is the density of Dirac operator eigenvalues.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 12 / 29 Banks-Casher relation Chiral symmetry breaking =⇒ =⇒ Low-lying eigenmodes of Dirac operator
Dirac equation: D[A] ψn = iλnψn, {γ5, γµ} = 0, D[A] γ5ψn = −iλnγ5ψn Non-zero eigenvalues appear in imaginary pairs ±iλn. 1 1 hψψ¯ i = − lim lim = m→0 V →∞ *V n m + iλn + X 1 1 1 1 = − lim lim dλ ρV (λ) + m→0 V →∞ V 2 m + iλ m − iλ Z m d m − lim = lim arctan −→ πδ(0) m→0 m2 + λ2 m→0 dλ λ Chiral condensate =⇒ Density of Near-Zero-modes πρ (0) hψψ¯ i = V V ➜ Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Banks,Feb. Casher, 2014 1980 13 / 29 Atiyah-Singer index theorem ◮ zero-modes of fermionic matrix: D[A]ψ(x)= 0 ◮ ψ has definite chirality: 1 ψR = (1 ± γ5)ψ, ⇒ γ5ψR = ±ψR L 2 L L ◮ Index theorem (wilson, overlap fermions):
n−, n+: number of left-/right-handed zeromodes
indD[A]= n− − n+ = Q[A] ◮ (Asqtad) staggered fermions: ind D[A]= 2Q[A] (SU(2), double degeneracy) ◮ Adjoint overlap fermions: ind D[A]= 2NQ[A]= 4Q[A] (real representation) ➜ Neuberger, Fukaya (1999)
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 14 / 29 Topological charge Q
◮ QCD-vacua characterised by winding number nw ◮ scalar (gauge) function: g(x)= e−i~α(x)~σ ∈ SU(2)≃ S3 ◮ 3 3 1 3 † † † R → S : nw = − 4π2 S3 d xSp(∂i gg ∂j gg ∂k gg ) ◮ † ~σ ~ vector field: i∂µgg =RAµ = 2 Aµ ◮ i i i j k Fµν = ∂µAν − ∂ν Aµ − εijk AµAν = 0
◮ 1 4 i ˜ i 1 ~ ~ Topological charge: Q = 32π2 d xFµν Fµν = 4π2 EB
◮ 1 † R Lattice: Fµν = 2i (Uµν − Uµν ) ◮ admissibility condition: tr(½ − Uµν ) < 0.0011 uniquness of topological charge on the lattice: ➜ Lüscher (1998)
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 15 / 29 Center vortices and topological charge
It is believed that in a non-Abelian gauge theory spontaneous chiral symmetry breaking is intimately linked to gauge configurations having topological charge. So one may ask whether center vortices can give rise to topological charge. Recall that the topological charge density is defined as 1 1 1 q(x)= Tr F F˜ = E~ · B~ , F˜ = ǫ F . 16π2 µν µν 4π2 µν 2 µνρσ ρσ So we need flux in all four directions. A vortex has flux perpendicular to its world sheet. Hence to generate topological charge we need two orthogonal vortices intersecting, or one vortex “writhing,” i.e., twisting around itself.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 16 / 29 Generating topological charge
Intersections and writhing points contribute to the topological charge of a P-vortex surface ◮ 1 intersections Q = ± 2 ◮ 1 writhing points Q = ± 4 H. Reinhardt, NPB628 (2002) 133 [hep-th/0112215], hep-th/0204194
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 17 / 29 Classical vortices and Dirac eigenmodes We can create a pair of plane vortices, or a vortex-antivortex pair, in the xy plane by setting the time like links in one time slice t0 to U4(t0, z)= exp{iσ3φ(z)} with φ as
2π 2π
2d 2d 2d φ1 π φ2 π 2d
0 0
1 z1 z z2 Nz 1 z1 z z2 Nz
All other links are set to unity, Uµ = ½. Due to periodic boundary conditions, plane vortices come in parallel or antiparallel pairs.
The corresponding P-vortices are located at z1 and z2.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 18 / 29 Intersecting plane vortices Intersecting two orthogonal pairs of plane vortices we can generate topology.A xy vortex generates a chromo-electric field, Ez , and a zt vortex a chromo-magnetic field, Bz . Each intersection point contributes Q = ±1/2 to the total topological charge.
Parallel Vortices Geometry Antiparallel Vortices z
+Q +Q 00 00 -Q xx y xx z z x
So we can get Q = 2 with parallel intersecting vortices and Q = 0 with antiparallel intersecting vortices.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 19 / 29 Intersecting plane vortices We define the scalar density as ρ = ψ†ψ and the chiral density as † ρ5 = ψ γ5ψ, with ψ an overlap Dirac operator eigenmode. Left: scalar density of the two zero modes for the Q = 2 intersecting vortex pairs on a 224 lattice. Right: chiral density of the near-zero modes of the Q = 0 intersecting vortex pairs on a 164 lattice.
y=8, t=8, chi=0, n=0-0, max=0.000429654 y=11, t=11, chi=-1, n=1-2, max=0.0000733948
0.0004 0.00006 0.0002 16 0.00004 20 0 0.00002 -0.0002 12 0 15 -0.0004 0 10 z 8 z 5 4 10 8 5 4 xx 15 xx 12 20 0 16
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 20 / 29 Planar vortices are “special”
The strictly planar (zy) vortices introduced, with the nontrivial gauge links U4(t0, z)= exp{iσ3φ(z)}, have all links in a single U(1) subgroup. They are thus like 2-d U(1) vortices, replicated in the two orthogonal directions (x and y). 2-d U(1) vortices have topological charge and thus exact zero modes. The replication in the orthogonal directions leaves this unchanged. And indeed, we find two zero modes for a parallel planar vortex pair, but no zero modes, with lifted near-zero modes instead, for an antiparallel planar vortex pair.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 21 / 29 Colorful spherical vortex
A nonorientable spherical vortex, with radius R and thickness ∆, is given by exp (iα(|~r −~r0|)~n · ~σ) t = ti ,µ = 4 Uµ(x)= ,
(½ elsewhere
with ~n = (~r −~r0)/|~r −~r0|, and ~r the spatial coordinates of site x. The profile function α is either
∆ 0 π r < R − 2 π r−R π r−R ∆ ∆ α+(r)= 2 1 − ∆ , α−(r)= 2 1 + ∆ R − 2 < r < R + 2 2 2 ∆ π 0 R + 2 < r
All other links are set to unity, Uµ = ½.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 22 / 29 Colorful spherical vortex The corresponding P-vortex is the sphere at radius R. The color structure of the spherical vortex, a hedgehog configuration, is illustrated in the left plot. The right plot illustrates the monopole lines after Abelian projection in the maximal Abelian gauge.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 23 / 29 Colorful spherical vortex Assigning to the space-like boundary constant U4(r)= ±1, 3 gives a map r ∈ S → SU(2) ∋ U4(r) characterized by a winding number
1 3 † † † N = d x ǫ Tr[(∂i U U )(∂j U U )(∂ U U )], 24π2 ijk 4 4 4 4 k 4 4 Z that contributes to the topology and leads to an exact zero mode.
0.2
z=16, t=1, chi=-1, n=0-0, max=0.0000822192
0.1 λ
0.00008 2 spherical vortex on 12 x6 lattice 0.00006 32
2 0.00004 spherical vortex on 16 x8 lattice 0.00002 24 spherical vortex on 242x12 lattice 0 16 y free overlap Dirac operator (trivial gauge field) 8 0 16 8 x 24 0 5 10 15 20 32 mode #
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 24 / 29 Dirac spectra, spherical vortices and instantons The overlap Dirac eigenvalues, and even the eigenmodes, in the background of spherical vortices are very similar to those with instantons.
0.25
0.2
0.15 λ 0.1
spherical SU(2) (anti-/)vortex 0.05 spherical vortex-anti-vortex pair single (anti-/)instanton instanton-anti-instanton pair instanton + anti-vortex free overlap Dirac operator (trivial gauge field) 0
0 4 8 12 16 mode #
With objects of opposite topological charge, the would-be zero modes interact and become near-zero modes. Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 25 / 29 Dirac spectra, spherical vortices and instantons Chiral density for the zero mode, first and ninth modes in an instanton background (top row) and a spherical vortex background (bottom row).
= = = - y 6, z 6, chi 0 6 y=6, z=6, chi=0, n=1-1, max=0.0000478351 , n=0-0, max=-4.0194 10 y=6, z=6, chi=0, n=9-9, max=0.0000496241
24 24 24 20 20 20 0.00004 16 0.00005 16 0 16 0.000025 -0.00005 0.00002 - 0 12 t 0 12 t 0.0001 12 t - -0.000025 -0.00015 0.00002 - -0.0002 -0.00004 0.00005 8 8 8 4 4 4 4 4 4 8 x 8 x 8 x 12 12 12
y=6, z=6, chi=0 -6 = = = = - = y=6, z=6, chi=0, n=9-9, max=0.0000582468 , n=0-0, max=-9.7466 10 y 6, z 6, chi 0, n 1 1, max 0.0000469721
24 24 24 20 20 20 0 16 0.00005 16 16 0.000025 0 - 0.00005 0 12 t -0.00005 12 t 12 t -0.000025 - -0.0001 - 0.0001 0.00005 8 8 8 4 4 4 4 4 4 8 8 8 x x x 12 12 12 Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 26 / 29 Dirac spectra, spherical vortices and instantons
The similarity of the behavior of the Dirac spectrum in the background of instantons and of spherical vortices suggests that almost classical vortices, similarly to an instanton liquid model, can produce a finite spectral density near zero and thus, by the Banks-Casher argument, lead to the spontaneous breaking of chiral symmetry.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 27 / 29 Summary
◮ Confirmed center dominance (saturation of string tension) with improved gauge action ◮ Confirmed that center vortices also induce chiral symmetry breaking ◮ So center vortices dominate long-distance, non-perturbative physics ◮ Classical vortices, planar and spherical, can contribute to the topological charge, leading to zero modes and near-zero modes ◮ A center vortex model of QCD can thus lead to both confinement and spontaneous chiral symmetry breaking
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 28 / 29 Vienna University of Technology
Thank you for your attention! Questions?
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 29 / 29 Notions of confinement There is another loop operator, B(C), introduced by ’t Hooft (1978), with C a loop on the dual lattice, intended to be “dual” to the Wilson loop operator, W (C ′), with the property
B(C)W (C ′)= exp(2πi/N)W (C ′)B(C) ,
if C and C ′ are topologically linked. Ina confined phase, B(C) has a perimeter law falloff, exp[−µP(C)], but an area law falloff otherwise. B(C) can be interpreted as creation operator of a “physical” center vortex, leading to the center vortex picture of confinement. Physical center vortices, with an expected thickness of order 1 fm, are rather difficult to locate in a gauge configuration. Much easier to define and find are “thin” vortices in center-projected gauge fields, called “P-vortices”. It is assumed that P-vortices track the physical center vortices.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 30 / 141 A puzzle
original (full) center-projected vortex-removed
0.1
λ 0 Im
-0.1
-0.05 0 0.05 -0.05 0 0.05 -0.05 0 0.05 Re λ Re λ Re λ
On full and vortex-removed configurations we observe the expected, but on center-projected configs, we find a gap and hence no χSB. Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 31 / 141 Center vortex dominance Well known with Wilson action. We test with Lüscher-Weisz action. Creutz ratios χ(R, R) approach string tension σ as R →∞.
β LW = 2.9
β LW = 3.1
β LW = 3.3 Lattice spacing a ≈ ) 0.1 0.27, 0.21, 0.15, 0.11 R,R
( β LW = 3.5 χ fm for βLW = 2.9, 3.1, 3.3, 3.5. asymptotic string tension L=84 L=124 L=164 L=204 0 1 2 3 4 5 6 7 8 R Find that center-projected Creutz ratios approach full string tension.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 32 / 141 Center vortex dominance
0.35 1.5 full theory center projected 0.3 vortex removed 1 asymptotic string tension 0.25 0.5 0.2 0
0.15 / W 0 n x(R,R) 0.1 W -0.5 0.05 -1 0 W1/W0 W /W -0.05 -1.5 2 0 0 1 2 3 4 5 6 0 5 10 15 20 25 R Loop Area
Left: Creutz ratios for full, center-projected, and vortex-removed gauge fields for βLW = 3.3.
Right: Wilson loop pierced by n P-vortices Wn. n Expect Wn → (−1) W0 as area is increased. Cancellations lead to area-law of confinement.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 33 / 141 Chiral symmetry breaking and Dirac spectrum Since center-projected gauge fields induce confinement, one expects them to induce chiral symmetry breaking. Vortex-removed, non-confining gauge fields are expected to have a gap and no χSB. Overlap fermions, with “exact chiral symmetry” should be a good probe.
aDov = M [1 + γ5ε (γ5DW (−M))] ,
satisfying the Ginsparg-Wilson relation a {γ , Dov } = Dov γ Dov , 5 M 5 or a γ , D−1 = γ . 5 ov M 5 n o This is the mildest breaking of continuum chiral symmetry on the lattice.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 34 / 141 Dirac spectra, spherical vortices and instantons
The overlap Dirac eigenvalues, and even the eigenmodes, in the background of spherical vortices are very similar to those with instantons.
0.25 0.2
0.2
0.15 0.15
λ λ 0.1 0.1
spherical SU(2) (anti-/)vortex 0.05 2 (anti-/)instantons 0.05 spherical vortex-anti-vortex pair 2 spherical SU(2) (anti-/)vortices single (anti-/)instanton 2 spherical vortex-anti-vortex pairs instanton-anti-instanton pair (anti-/)vortex + vortex-anti-vortex pair instanton + anti-vortex free overlap Dirac operator (trivial gauge field) free overlap Dirac operator (trivial gauge field) 0 0
0 4 8 12 16 0 4 8 12 16 mode # mode #
With objects of opposite topological charge, the would-be zero modes interact and become near-zero modes.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 35 / 141 Dirac spectra, spherical vortices and instantons Chiral density for the first (near-zero mode), second and ninth modes in an instanton–anti-instanton (top row) and a spherical vortex-antivortex background (bottom row).
y=6, z=6, chi=0, n=0-0, max=0.00212786 y=6, z=6, chi=0, n=1-1, max=0.000043833 y=6, z=6, chi=0, n=9-9, max=0.0000617173
24 24 24 20 20 20 0.0005 16 0.00004 16 0.00005 16 0.00025 0.00002 0 12 t 0 12 t 0 12 t -0.00025 -0.00002 - - -0.00005 0.0005 8 0.00004 8 8
4 4 4 4 4 4 x 8 x 8 x 8 12 12 12
= = = = - = y 6, z 6, chi 0, n 0 0, max 0.0012622 y=6, z=6, chi=0, n=1-1, max=0.000055155 y=6, z=6, chi=-1, n=9-9, max = 0.0000998798
24 24 24 20 20 20 16 0.00005 16 0.0001 16 0.0002 0.000025 0.000075 0 12 t 0 12 t 0.00005 12 t - -0.000025 0.000025 0.0002 -0.00005 0 8 8 8 4 4 Manfried Faber4 (Atominstitut)4 Vortices and chiral symmetry4 breaking Feb.4 2014 36 / 141 8 x 8 x 8 x 12 12 A puzzle Confirm Gattnar et al., NPB716 (2005) 105 [hep-lat/0412032], who used a “chirally improved” Dirac operator.
original center projected original vortex-removed 0.6 0.4
0.4 0.2 0.2 λ λ 0.0 0.0 Im Im
-0.2 -0.2 -0.4
-0.6 -0.4 0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.2 0.4 Re λ Re λ Re λ Re λ
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 37 / 141 A puzzle Compare to improved staggered (asqtad) fermions:
original (full) center-projected vortex-removed
0.1 0.1 0.1 λ
Im 0 0 0
-0.1 -0.1 -0.1
Here find expected behavior on center-projected configs, too.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 38 / 141 And its resolution
The chiral symmetry of overlap fermions is not quite the usual one a δψ = iαγ 1 − D ψ , δψ¯ = iαψγ¯ . 5 M ov 5 It is field dependent for finite a.
For center-projected configs U˜µ(x)= ±1. So the configs are very rough, and easily Dov (U˜ )= O(1/a), i.e., aDov (U˜ )= O(1). Then the notion of “exact chiral symmetry” is lost. Note that staggered fermions have a field-independent U(1) × U(1) chiral symmetry. So center-projected configs induce no gap, signaling χSB, as expected.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 39 / 141 And its resolution
Interpolating, on the SU(2) manifold, between Uµ(x) and U˜ µ(x), the configs can be made smoother. For sufficiently smooth interpolation, that gap closes.
original (full) 50% projected 75% projected 85% projected 90% projected 95% projected center-projected
0.1 λ 0 Im
-0.1
0 0 0 0 0 0 0 Re λ Re λ Re λ Re λ Re λ Re λ Re λ
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 40 / 141 Intersecting plane vortices We compare a few low-lying overlap eigenvalues in plane vortex configurations with the free case, always using antiperiodic boundary conditions in the time direction.
free overlap Dirac operator (trivial gauge field) 0.3 plane vortex pair with opposite flux (anti-parallel) crossing of anti-parallel vortex pairs (Q=0) plane vortex pair with same flux (parallel) 0.25 crossing of parallel vortex pairs (Q=2)
0.2
λ 0.15
0.1
0.05
0
0 8 16 mode #
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 41 / 141 Confinement due to Magnetic Monopoles? type II superconductor dual superconductor
magnetic fluxoid quantisation electric fluxoid quantisation
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 42 / 141 Monopoles versus Vortices
The “spaghetti vacuum” appears, under abelian projection,
+
+ - + - - + + + -
-
+
- -
+
as a “monopole vacuum”
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 43 / 141 Monopoles and Vortices
Piercing of monopole cubes by vortices
➜ Greensite et al. (1997)
3 % 93 % 4 % No vortex 1 vortex >1 vortex
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 44 / 141 ’t Hooft loop
t t t
z z z
How to create a ’t Hooft loop ∂Σ˜ in the (x, y) plane: (left) in each (z, t) plane intersecting Σ˜, multiply by a center element one plaquette with coordinates (z0, t0); (middle) equivalently, choose (z0, t0) in the corner; (right) equivalently, multiply by a center element the link Ut at the boundary. Thus, a ’t Hooft loop of maximal size is equivalent to twisted boundary conditions for the Polyakov loop.
1 1 DU exp(−β (1 − ReTr ζUP ) − β (1 − ReTr UP )) U ∈P(Σ)˜ N U ∈P/ (Σ)˜ N hW˜ (∂Σ)˜ i ≡ P P . 1 ReTr DU exp(−β (1 − N UP )) R P UP P
Manfried Faber (Atominstitut) VorticesR and chiral symmetryP breaking Feb. 2014 45 / 141 Center vortex sketch
y
z z z z z z z x y0
x 0
Left: Creation of a thin center vortex: x − y plaquettes (shaded) at sites (x0, y0,~x⊥) form the center vortex (on the dual lattice). Right: Three center vortices (solid lines), for SU(3), piercing plaquettes on the original lattice (dashed contours) diverging from a monopole (solid circle) on the dual lattice.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 46 / 141 Dirac spectrum at finite temperature
123 x 2 123 x 4 123 x 6 124 1.2 1.2 1.2 1.2 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 λ 0 0 0 0 Im -0.2 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 -0.8 -1 -1 -1 -1 -1.2 -1.2 -1.2 -1.2
Finite temperature and center projection: 3 Asqtad Dirac-operator eigenvalues on 12 × Nt center-projected lattices with Nt = 2, 4, 6, 12 at βLW = 3.5 – above βc (Nt = 6). On center-projected lattices chiral restoration appears to occur at higher temperatures than deconfinement. Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 47 / 141 Vortices and Dirac eigenmode densities
On center-projected configs, Dirac eigenmodes can be expected to be correlated with center vortices. Does this hold for red full configs? Note: center vortices generate topological charge at intersection and “writhing” points (Engelhardt and Reinhardt (2000)) Define correlator ( see Kovalenko et al., PLB648 (2007) 383 [hep-lat/0512036])
pi x∈H(V ρλ(x) − 1) Cλ(Nv)= , P P pi x∈H 1
2 P P where ρλ(x)= |ψλ(x)| , pi points on the dual lattice with Nv plaquettes on the vortex surface, and H the hypercube on the original lattice dual to the point pi .
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 48 / 141 Vortex correlation: staggered fermions
The vortex correlation Cλ(Nv) as function of the number of vortex plaquettes Nv at a given site.
1.5 5 1st mode 1st mode 20th mode 20th mode
4
1
3
0.5 2
1 vortex correlation 0 vortex correlation
0
-0.5 -1 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 number of attached plaquettes number of attached plaquettes Full configs Center-projected configs
The eigenmode density is larger on sites with larger Nv, and more pronounced on center-projected configs, where center vortices are thin and extremely localized.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 49 / 141 Vortex correlation: overlap fermions
1.2 0.2 zero mode 1st mode 1st mode 20th mode 20th mode 1 0.1
0 0.8
-0.1 0.6 -0.2 0.4 -0.3
0.2 vortex correlation vortex correlation -0.4
0 -0.5
-0.2 -0.6 0 2 4 6 8 10 0 2 4 6 8 10 12 14 number of attached plaquettes number of attached plaquettes Full configs Center-projected configs
For full configs, vortex correlation Cλ(Nv) for overlap fermions is similar to staggered fermions. For center-projected – and too rough for overlap fermions – we see even an anti-correlation.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 50 / 141 Dimensionality of Dirac eigenmodes densities
How localized are the low-lying eigenmodes? Often used to quantify localization of eigenmodes is the inverse participation ratio (IPR)
2 I = N ρλ(x) . x X If an eigenmode has support on a submanifold of dimension d, with thickness in the orthogonal 4 − d direction a fixed number of lattice spacings, then I ∝ 1/a4−d . The results are controversial, ranging from d = 0 (pointlike) to d = 3. Scaling of the number of “disconnected” support structures can change the scaling behavior of I.
We just inspect by looking at ρλ(x):
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 51 / 141 Dimensionality: staggered fermions
z=10, t=14 z=11, t=14 z=2, t=7 z=3, t=7
0.0006 0.0006 0.0076 0.0076 0.0005 0.0005 0.0004 0.0004 0.0057 0.0057 0.0003 0.0003 0.0038 0.0038 0.0002 0.0002 0.0001 20 0.0001 20 0.0019 20 0.0019 20 16 16 16 16 0 20 12 0 20 12 0 20 12 0 20 12 16 16 16 16 12 8 y 12 8 y 12 8 y 12 8 y 8 8 8 8 x 4 4 x 4 4 x 4 4 x 4 4
density of eigenvalue #1, maximum 0.000553522856787 at x=5, y=9, z=12, t=14 density of eigenvalue #1, maximum 0.00745768618163 at x=9, y=8, z=4, t=7
0.0006 0.0076 0.0005
0.0057 0.0004
0.0003 0.0038
0.0002 20 0.0019 20 0.0001 16 16
0 20 12 0 20 12 16 16 8 y 8 y 12 12 8 4 8 4 x 4 x 4 0 0 0 0
z=13, t=14 z=14, t=14 z=5, t=7 z=6, t=7
0.0006 0.0006 0.0076 0.0076 0.0005 0.0005 0.0004 0.0004 0.0057 0.0057 0.0003 0.0003 0.0038 0.0038 0.0002 0.0002 0.0001 20 0.0001 20 0.0019 20 0.0019 20 16 16 16 16 0 20 12 0 20 12 0 20 12 0 20 12 16 16 16 16 12 8 y 12 8 y 12 8 y 12 8 y 8 8 8 8 x 4 4 x 4 4 x 4 4 x 4 4
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 52 / 141 Dimensionality: overlap fermions
z=3, t=2 z=4, t=2 z=13, t=3 z=14, t=3
0.0001 0.0001 0.0032 0.0032 8e-05 8e-05 0.0024 0.0024 6e-05 6e-05 0.0016 0.0016 4e-05 4e-05 0.0008 16 0.0008 16 2e-05 16 2e-05 16 12 12 12 12 0 16 0 16 0 16 0 16 12 8 y 12 8 y 12 8 y 12 8 y 8 4 8 4 8 4 8 4 x 4 x 4 x 4 x 4
density of eigenvalue #1, maximum 0.0032086919732 at x=11, y=12, z=5, t=2 density of eigenvalue #1, maximum 8.84376963119e-05 at x=9, y=10, z=15, t=3
0.0001 0.0032 8e-05 0.0024 6e-05
0.0016 4e-05
16 16 0.0008 2e-05 12 12 0 16 0 16 8 8 12 y 12 y 8 4 8 4 4 4 x x 0 0 0 0
z=6, t=2 z=7, t=2 z=0, t=3 z=1, t=3
0.0001 0.0001 0.0032 0.0032 8e-05 8e-05 0.0024 0.0024 6e-05 6e-05 0.0016 0.0016 4e-05 4e-05 0.0008 16 0.0008 16 2e-05 16 2e-05 16 12 12 12 12 0 16 0 16 0 16 0 16 12 8 y 12 8 y 12 8 y 12 8 y 8 4 8 4 8 4 8 4 x 4 x 4 x 4 x 4
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 53 / 141 Topological charge Q
◮ QCD-vacua characterised by winding number nw ◮ scalar (gauge) function: g(x)= e−i~α(x)~σ ∈ SU(2)≃ S3
◮ 3 3 1 3 † † † R → S : nw = − 4π2 S3 d xSp(∂i gg ∂j gg ∂k gg )
◮ R 1 4 i ˜ i 1 ~ ~ Topological charge: Q = 32π2 d xFµν Fµν = 4π2 EB R ◮ 1 † Lattice: Fµν ⇒ 2i (Uµν − Uµν ) ◮ admissibility condition: tr(½ − Uµν ) < 0.0011 uniquness of topological charge on the lattice
➜ Lüscher (1998)
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 54 / 141 Topological charge Q
◮ QCD-vacua characterised by winding number nw ◮ scalar (gauge) function: g(x)= e−i~α(x)~σ ∈ SU(2)≃ S3 ◮ † ~σ ~ vector field: i∂µgg = Aµ = 2 Aµ ◮ ′ ′ perturbative vacuum: Aµ(x) ≡ 0, nw = 0 ′ † † † Aµ(x)= g(Aµ − i∂µ)g = −ig∂µg = i∂µgg ◮ 3 3 1 3 † † † R → S : nw = − 4π2 S3 d xSp(∂i gg ∂j gg ∂k gg ) ◮ i i i j k Fµν = ∂µAν − ∂ν Aµ −Rεijk AµAν = 0
◮ ′ vacuum to vacuum transition: Q = nw − nw ◮ 1 4 i ∗ i Topological charge: Q = 32π2 d xFµν Fµν
◮ 1 † R Lattice: Fµν = 2i (Uµν − Uµν ) ◮ admissibility condition: tr(½ − Uµν ) < 0.03 uniquness of topological charge on the lattice: ➜ Lüscher (1998)
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 55 / 141 Atiyah-Singer index theorem ◮ zero-modes of fermionic matrix: D[A]ψ(x)= 0 ◮ ψ has definite chirality: 1 ψR = (1 ± γ5)ψ, ⇒ γ5ψR = ±ψR L 2 L L ◮ Index theorem (wilson, overlap fermions):
n−, n+: number of left-/right-handed zeromodes
indD[A]= n− − n+ = Q[A] ◮ (Asqtad) staggered fermions: ind D[A]= 2Q[A] (SU(2), double degeneracy) ◮ Adjoint overlap fermions: ind D[A]= 2NQ[A]= 4Q[A] (real representation) ➜ Neuberger, Fukaya (1999)
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 56 / 141 Questions
◮ Do zero-modes locate topological charge distributions? † † analyse ρ(x)= ψ (x)ψ(x), ρ5(x)= ψ (x)γ5ψ(x)
◮ Are there configurations with fractional topological charge? 1 i ∗ i analyse Q(x)= 32π2 Fµν(x) Fµν(x)
◮ Can we identify fractional topological charge with adjoint fermions? (Pisarsky, Greensite)
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 57 / 141 Topological excitations
◮ instantons: vacuum-vacuum-transitions ◮ merons, calorons, dyons: parts of instantons ◮ vortices: singular gauge transformations, world sheets of magnetic flux ◮ magnetic monopoles: projections of magnetic flux or singularities of the gauge field
➜ Ilgenfritz, Martemyanov, Müller-Preussker, Shcheredin, Veselov, Phys.Rev.2002
➜ Ilgenfritz, Müller-Preussker, Peschka, Phys.Rev.2005
➜ Ilgenfritz, Martemyanov, Müller-Preussker, Veselov, Phys.Atom.Nucl.2005
➜ Ilgenfritz, Martemyanov, M. Müller-Preussker, Veselov, Phys.Rev.2006
➜ Bornyakov, Ilgenfritz, Martemyanov, Morozov, M. Müller-Preussker, Veselov, Phys.Rev.2007
➜ Bornyakov, Ilgenfritz, Martemyanov, Morozov, Müller-Preussker, Veselov, Phys.Rev.2008
➜ Bornyakov, Ilgenfritz, Martemyanov, M. Müller-Preussker, Phys.Rev.2009
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 58 / 141 Center Vortices
➜ ’t Hooft 1979, Nielsen, Ambjorn, Olesen, Cornwall, 1979 Mack, 1980; Feynman, 1981
◮ QCD vacuum is a condensate of closed magnetic flux-lines, they have topology of tubes (3D) or surfaces (4D), ◮ magnetic flux corresponds to the center of the group, ◮ Vortex model may explain ... ◮ Confinement → piercing of Wilson loop ≡ crossing of static electric flux tube and moving closed magnetic flux ◮ Topological charge: vortices carry topological charge at intersection points and writhing points ◮ Spontaneous chiral symmetry breaking: also center-projected configurations show SCSB
➜ Höllwieser, M.F., Greensite, Heller, Olejnik 2008
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 59 / 141 Definition of Plain Vortices
xy-vortices: along z-direction we vary t-links, in a U(1) subgroup of SU(2), e.g. σ3, Uµ = exp{iφ(z)σ3}.
2π 2π
2d 2d 2d φ1π φ2π 2d
0 0
a) 1 z1 z z2 Nz b) 1 z1 z z2 Nz
a) φ1 of a parallel and b) φ2 of an anti-parallel vortex pair
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 60 / 141 Thick Spherical SU(2)-vortices
Phase
pi
pi/2
0
0 5 10 15 20 25 30 35 40 x ~r ν exp{iα(r) ~σ}, t = 1,µ = 4 Uµ(x )= r 1 else
π ∆ 0 r < R − 2 α (r)= nπ 1 ∓ r−R R − ∆ < r < R + ∆ ± 2 ∆ 2 2 2 0 ∆ π R + 2 < r Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 61 / 141 n Topological charge of intersecting vortices
xy-vortices: along z-direction we vary t-links, zt-vortices: along x-direction the y-links vary ⇒ x-z-intersection plane Parallel Vortices Geometry Anti-parallel Vortices z
++Q ++QQ 00 −−Q 00 y xx xx z z x
Q=2 Q=0
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 62 / 141 Contributions to topological charge Q
vortex intersection
4 · 4 = 16 contributions 1 contribution
1 1 Q = ± 2 Q = ± 32
vortex surfaces need orientation Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking➜ Engelhardt, ReinhardtFeb. 2014 (2000) 63 / 141 Intersections and Writhing points
contributions to topological charge ◮ 1 intersections Q = ± 2 ◮ 1 writhing points Q = ± 4
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 64 / 141 Analytical results
➜ Reinhardt, Schroeder, Tok and Zhukovsky (2002)
Analytical calculations by the Tübingen group. Zero-modes peak at intersections of vortices.
30
20 1
10 0.8
0 0.6 0 0.2 0.4 0.4 0.6 0.2 0.8 1 0
Probability density of zero-mode in the background of four intersecting vortices. Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 65 / 141 Zero-modes for Q=2 y=11, t=11, chi=-1, n=1-2, max=0.0000498734
periodic boundary conditions
0.00005 0.00004 0.00003 0.00002 20 0.00001 15 0 0 10 z 5 2 10 5 x 15 1 20 0 0 -1 20 y=11, t=11, chi=-1, n=1-2, max=0.0000733948 -2 15 0 5 10 z 10 5 x 15 20 0
0.00006 Sp Ly + Sp Lt 0.00004 20 0.00002 0 15 antiperiodic boundary cond. 0 10 z 5 intersections at x and z = 5.5 and 16.5 10 5 Manfried Faber (Atominstitut)x 15 Vortices and chiral symmetry breaking Feb. 2014 66 / 141 20 0 Perpendicular to intersections
x=5, z=5, chi=-1, n=1-2, max=0.0000503624
0.00005 0.00004 0.00003 20 0.00002 0.00001 15 0 0 10 t 5 10 5 y 15 20 0 zero-modes at maximum, perpendicular to intersections
Conclusions: ◮ Vortices contribute with fractions of topological charge ◮ Zero-modes are related to vortex intersections ◮ Zero mode positions depend on values of Polyakov loops
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 67 / 141 Localisation 2 Scalar density: ρi (x)= c,d |ψi (x)cd | c and d, color und Dirac indices P N x=1 ρi (x)= 1 P c,d′ ∗ 1−γ5 Chiral densities ρ+(x) and ρ−(x): ρi±(x)= c,d ψi (x)cd 2 ψi (x)cd′ P Inverse participation ratio (IPR)
➜ Ilgenfritz:2007, Polikarpov:2005, Aubin:2004, Bernard:2005
1 I = fraction of contributing sites
N 2 1 I = N x=1 ρi (x), N ≤ I ≤ N
locateP fractional topological charge contributions with I
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 68 / 141 Scalar densities for adjoint overlap fermions Adjoint fermions: indD[A]= 2NQ[A]= 4Q[A] hope to separate various contributions
Q=0 configuration with periodic boundary conditions: no zero-modes for fundamental fermions 2+2 zero-modes for adjoint fermions
= = =- = = = y=11, t=11, chi=-1, y 12, t 12, chi 1, y 11, t 11, chi 1, = - = −6 = - = −6 n=1-2, max=8.5946 10−6 n 1 2, max 8.601 10 n 1 2, max 8.5714 10
20 20 20 15 15 15 0 10 0 10z 0 10 5 z 5 5 z 10 5 10 5 10 5 xx 15 xx 15 xx 15 a) 20 0 b) 20 0 c) 20 0 negative positive chirality
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 69 / 141 Adjoint overlap fermions
Q=2 configuration with periodic boundary conditions: Scalar density plots 2 positive and 10 negative adjoint zero-modes
y=8, t=8, chi=-1, n=1-10, max=0.00140884 y=8, t=8, chi=-1, n=1-2, max=0.00003054
0.00003054 0.00125 0.00003052 15 0.001 15 0.0000305 0.00075 0.0005 0.00003048 10 10 0 z 0 z 5 5 5 5 10 xx 10 15 0 xx 0 a) b) 15 2 non-topological 10 negative chirality zero-modes
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 70 / 141 Search for IPR maxima Q=2 configuration with antiperiodic boundary conditions: parameter space of linear combinations of the eight topological zero-modes, systematic and random (20 000) search of maxima
0.0008 0.0007 1.1 0.0006 1 0.0005 0.9 0.0004 0.8 0.0003 IPR 0.7 0.0002 0 0.6 0.0001 4 0.5 0 0 4 8 8 z 12 x 12 16 16 a) b) a) Plane through the highest three IPR maxima: The peaks are very broad and easy to identify b) Density with maximal IPR still peaks at two vortex Manfriedintersections Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 71 / 141 Q = 1-configuration π2
d φ 3
TrUA 0 −1 xx
a) 1 z Nz b) z
Thin and thick vortices: xy-vortices: along z-direction we vary t-links, zt-vortices: along x-direction the y-links vary Adjoint fermions are blind against thin vortices: † 2 (TrU)(TrU) = (TrU) = 1 + TrUA a) Link profile b) Topological charge density in the intersection plane, "thick-thick" vortex intersection Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 72 / 141 Q = 1 2-configuration Adjoint fermions: indD[A]= 2NQ[A]= 4Q[A] xy-vortices: along z-direction we vary t-links, zt-vortices: along x-direction the y-links vary antiperiodic boundary conditions two adjoint overlap zero-modes
y=11, t=11, chi=-1, n=1-2, max=0.000070289 x=11, z=11, chi=-1, n=1-2, max=0.0000192506 PA 3 1 −1 0.00006 0.000015 −3 0.00004 20 0.00001 20 - 0.00002 5 ´ 10 6 15 15 0 0 0 0 10 z 10 t 5 5 10 5 10 5 xx 15 yy 15 20 0 a) 20 0 b) Zero-modes avoid negative adjoint Polyakov (Wilson) lines
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 73 / 141 Q = 1 2-configuration xy-vortices: along z-direction we vary t-links, zt-vortices: along x-direction the y-links vary antiperiodic boundary conditions
x=11, y=11, chi=-1, z=11, t=11, chi=-1, n=1-2, max=0.000070289 = - = −7 n 1 2, max 9.736 10 P PA A 3 1
−1 −3 0.00006 - 7.5 ´ 10 7 0.00004 - 20 5 ´ 10 7 20 0.00002 - 2.5 ´ 10 7 15 15 0 0 0 10 y 0 10 t 5 5 10 5 10 5 xx 15 zz 15 a) 20 0 b) 20 0 a) in x-direction profile of adjoint y-Wilson lines, b) in z-direction profile of adjoint t-Wilson lines
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 74 / 141 Q = 1 2-configuration various vortex profiles xy-vortices: along z-direction we vary t-links, zt-vortices: along x-direction the y-links vary antiperiodic boundary conditions y=11, t=11, chi=-1, n=1-2, max=0.0000706566 y=11, t=11, chi=-1, n=1-2, max=0.000084769 PA PA 3 3 1 1 −1 −1 0.00006 0.00008 −3 0.00006 −3 0.00004 20 0.00004 20 0.00002 0.00002 15 15 0 0 0 10 z 0 10 z 5 5 10 5 10 5 xx 15 xx 15 a) 20 0 b) 20 0 dx = dz = 16 dx = dz = 12 Zero mode densities follow values of adjoint Wilson lines
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 75 / 141 Q = 1 2-configuration various vortex profiles xy-vortices: along z-direction we vary t-links, zt-vortices: along x-direction the y-links vary antiperiodic boundary conditions y=11, t=11, chi=-1, n=1-2, max=0.000107012 y=11, t=11, chi=-1, n=1-2, max=0.000163089 PA PA 3 3 1 1
0.0001 −1 0.00015 −1 0.000075 −3 0.0001 −3 0.00005 20 20 0.000025 0.00005 15 15 0 0 0 10 z 0 10 z 5 5 10 5 10 5 xx 15 xx 15 20 a) 0 b) 20 0 dx = dz = 8 dx = dz = 4 Zero mode densities follow values of adjoint Wilson lines
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 76 / 141 Q = 1 2-configuration various vortex profiles xy-vortices: along z-direction we vary t-links, zt-vortices: along x-direction the y-links vary antiperiodic boundary conditions y=11,t=11,chi=-1, n=1-4, max=0.000723127 y=11,t=11,chi=-1, n=1-4, max=0.000918561
PA PA 3 3 1 1 −1 −1 0.0006 0.0006 −3 −3 0.0004 20 0.0004 20 0.0002 0.0002 15 15 0 0 0 10 z 0 10 z 5 5 10 5 10 5 xx 15 xx 15 a) 20 0 b) 20 0 dx = 20, dz = 8 dx = 8, dz = 2 Zero mode densities follow values of adjoint Wilson lines and cover the whole lattice extent
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 77 / 141 Thick Spherical SU(2)-vortices, 403 ∗ 2-lattice
Phase
pi
pi/2
0
0 5 10 15 20 25 30 35 40 x
~r ν exp{iα(r) ~σ}, t = 1,µ = 4 Uµ(x )= r 1 else
~r L(~r)= exp{iα(r) ~σ} time-like Wilson lines r 1 admissibility condition is fullfilled: 1 − 2 trU ≤ 0.015 Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 78 / 141 Non-orientable spherical vortices
Non-orientable vortex surface . . . leads to monopole lines after abelian projection x yy
zz
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 79 / 141 Thick spherical SU(2)-vortex, 403 ∗ 2-lattice “Positive” Spherical Vortex y=7, t=1, chi=0, n=1-4, max=0.0000220383
Phase
0.00003 pi 0.00002 40
0.00001 30 pi/2 0 20 z 10 0 20 10 x 30 0 5 10 15 20 25 30 35 40 40 x ◮ Index of overlap operator:
n+ = 3, n− = 4 =⇒ indD[A]= n− − n+ = 1
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 80 / 141 Two thick spherical SU(2)-vortices, 403 ∗ 2-lattice One vortex at t = 1 and another at t = 2
y=7, t=1, chi=0, n=1-4, max=0.0000314321 y=7, t=1, chi=0, n=5-5, max=0.00315628
0.00003 0.00002 40 0.003 0.00001 30 0.002 40 0 0.001 20 30 10 z 0 20 10 20 z x 30 10 40 20 10 x 30 40 Polykov loop 2π
π Index of overlap operator: 0 n+ = 3, n− = 5 0 5 10 15 20 25 30 35 40 x indD[A]= n− − n+ = 2 Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 81 / 141 Thick spherical SU(2)-vortex, 403 ∗ 2-lattice “Negative” Spherical Vortex
y=7, t=1, chi=0, n=1-1, max=0.0014006
Phase 0.0015 pi 0.001 40 0.0005 30 pi/2 0 20z 10 20 10 0 x 30 40 0 5 10 15 20 25 30 35 40 x ◮ Index of overlap operator:
n+ = 1, n− = 0 =⇒ indD[A]= n− − n+ = −1
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 82 / 141 Two thick spherical SU(2)-vortices, 403 ∗ 2-lattice One vortex at t = 1 and another at t = 2 Phase Phase
pi pi
pi/2 pi/2
0 0
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 x x Index of overlap operator: n+ = 0, n− = 0 ⇒ indD[A]= n− − n+ = 0 For all examples of spherical vortices: ◮ only time-like links non-trivial ⇒ no magnetic field 1 4 i ∗ i ⇒ B = 0 ⇒ Q[A] = 32π2 d xFµν Fµν = 0! ◮ 1 admissibility condition is fullfilled: 1R− 2 trU ≤ 0.015 NO save determination of indD[A] with Q[A]!
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 83 / 141 Cooling histories of spherical vortex
◮ Index of the overlap operator: ind D = 1 ◮ Topological charge before cooling: B~ = 0 ⇒ Q[A]= 0 ◮ Topological charge with cooling: Q[A] ≈ 1
10 action S 1 10 action S 1 topological charge Q topological charge Q 8 0.8 8 0.8
6 0.6 6 0.6 inst inst Q Q
S/S 4 0.4 S/S 4 0.4
2 0.2 2 0.2
0 0 0 0
0 100 200 300 400 500 600 0 50 100 150 200 cooling steps cooling steps 403 × 4-lattice 403 × 2-lattice
What is the object appearing after 120 cooling steps?
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 84 / 141 Cooling a spherical vortex on a 403 × 2 lattice
0 cooling steps 20 cooling steps 40 cooling steps
60 cooling steps 70 cooling steps 78 cooling steps
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 85 / 141 Monopole loops during cooling
10 cooling steps 30 20 cooling steps 30 cooling steps 40 cooling steps 50 cooling steps 60 cooling steps 70 cooling steps 25 80 cooling steps 90 cooling steps
20
15
10 10 15 20 25 30 35 40 After 100 cooling steps all Abelian monopole loops are gone Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 86 / 141 Profile 1 functions of the Polyakov loop
0.8
0.6
0.4
0.2
0 Polyakov -0.2
-0.4 0 cooling steps -0.6 30 cooling steps 60 cooling steps -0.8 90 cooling steps 120 cooling steps 150 cooling steps -1 0 5 10 15 20 25 30 35 40 (x,20,20)
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 87 / 141 Polyakov loop components during cooling 1 Trace of Polyakov line at (20,20,20) 1,2,3-components of Polyakov line 0.8
0.6
0.4
0.2
0 Polyakov -0.2
-0.4
-0.6
-0.8
-1 0 20 40 60 80 100 120 140 160 Manfried Faber (Atominstitut) Vortices and chiralcooling symmetry steps breaking Feb. 2014 88 / 141 Central cube after 120 cooling steps
z
y
(20,20,20) x
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 89 / 141 Color flux through central plaquettes z
y
(20,20,20) x
Central cube is a Dirac monopole in non-Abelian representation with a gauge field fading away at large distances Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 90 / 141 Overlap, Asqtad & Adjoint Results
negative spherical vortex fundamental: adjoint: lattice: ovl: apbc: stag: apbc: ovl: apbc: stag: apbc: 84: 3+ 4- 0+ 1- 6+ 8- 0+ 2- 4+ 6- 0+ 2- 8+ 12- 0+ 4- 124: 3+ 4- 0+ 1- 6+ 8- 0+ 2- 4+ 6- 0+ 2- 8+ 12- 0+ 4- 164: 3+ 4- 0+ 1- 6+ 8- 0+ 2- 4+ 8- 0+ 4- 8+ 16- 0+ 8- 204: 3+ 4- 0+ 1- 6+ 8- 0+ 2- 4+ 8- 0+ 4- 8+ 16- 0+ 8- 403 × 2: 3+ 4- 0+ 1- 6+ 8- 0+ 2- 4+ 8- 0+ 4- 8+ 16- 0+ 8- cooled: 3+ 4- 0+ 1- 6+ 8- 0+ 2- 4+ 6- 0+ 2- 8+ 12- 0+ 4- 403 × 4: 3+ 4- 0+ 1- 6+ 8- 0+ 2- 4+ 8- 0+ 4- 8+ 16- 0+ 8-
positive spherical vortex fundamental: adjoint: lattice: ovl: apbc: stag: apbc: ovl: apbc: stag: apbc: 84: 1+ 0- 4+ 3- 2+ 0- 8+ 6- 6+ 4- 2+ 0- 12+ 8- 4+ 0- 124: 1+ 0- 4+ 3- 2+ 0- 8+ 6- 6+ 4- 2+ 0- 12+ 8- 4+ 0- 164: 1+ 0- 4+ 3- 2+ 0- 8+ 6- 8+ 4- 4+ 0- 16+ 8- 8+ 0- 204: 1+ 0- 4+ 3- 2+ 0- 8+ 6- 8+ 4- 4+ 0- 16+ 8- 8+ 0- 403 × 2: 1+ 0- 4+ 3- 2+ 0- 8+ 6- 8+ 4- 4+ 0- 16+ 8- 8+ 0- cooled: 1+ 0- 4+ 3- 2+ 0- 8+ 6- 6+ 4- 2+ 0- 12+ 8- 4+ 0- 403 × 4: 1+ 0- 4+ 3- 2+ 0- 8+ 6- 8+ 4- 4+ 0- 16+ 8- 8+ 0-
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 91 / 141 Answers
◮ Vortices contribute with fractions to topological charge ◮ Number of zero-modes is related to vortex intersections ◮ Zero mode positions depend on values of Wilson lines ◮ 1 Adjoint zero-modes can’t be attributed to Q = 2 configurations ◮ Even for “admissible” fields: ind D[A] 6= Q[A] ◮ Dirac monopoles differ from Abelian projected monopoles ◮ Dirac monopoles fading away at large distances have 1 Q = 2
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 92 / 141 Chiral Symmetry Breaking ◮ parity acting on a Dirac fermion is called chiral symmetry ◮ two chiralities of quarkfield ψ
1 ¯ † ¯1 ψL = (1 ± γ5)ψ, ψL = ψL γ0 = ψ (1 ∓ γ5) R 2 R R 2
2 † γ5 = iγ0γ1γ2γ3, γ5 = 1, γ5 = γ5, {γµ, γ5} = 0 ◮ 1 chiral projection PL = (1 ∓ γ5) on QCD-Lagrangian: R 2 µ ◮ kinetic term ψγ¯ Dµψ gives
ψγ¯ µψ = ψ¯R γµψR + ψ¯LγµψL
◮ whereas mass term mψψ¯ gives interaction
ψψ¯ = ψ¯R ψL + ψ¯LψR and breaks chiral symmetry explicitly ◮ spontanous symmetry breaking gives rise to Goldstone-boson: pion
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 93 / 141 Vortex removal restores chiral symmetry
➜ De Forcrand and D’Elia (1999)
Chiral condensate in quenched lattice configurations before (“Original”) and after (“Modified”) vortex removal.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 94 / 141 Banks-Casher relation Chiral symmetry breaking =⇒ =⇒ Low-lying eigenmodes of Dirac operator
1 1 ψψ¯ = − lim lim m→ V →∞ 0 *V n iλn + m + X Non-zero eigenvalues appear in pairs ±iλn 2m lim 2 2 −→ πδ(0) m→0 λn + m Chiral condensate =⇒ Density of Near-Zero-modes.
πρ(0) ψψ¯ = V ➜ Banks, Casher, 1980
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 95 / 141 Chiral Improved Fermions
original vortex-removed center projected 0.4
0.2 λ 0.0 Im
-0.2
-0.4 0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.2 0.4 Re λ Re λ Re λ
➜ From J. Gattnar et al., Nucl. Phys. B716 (2005)105.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 96 / 141 Eigenvalues of the Overlap Dirac operator on the Ginsparg-Wilson circle
original (full) center-projectedcenter projected vortex-removedvortex removed
0.1
λ 0 Im
-0.1
-0.05 0 0.05 -0.05 0 0.05 -0.05 0 0.05 Reλ Reλ Reλ
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 97 / 141 Interpolated gauge fields P Interpolation between full Uµ and center-projected U0 = ±1. x0
xj Uµ P U0 α
xi
P U0 = −1
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 98 / 141 Overlap eigenvalues on interpolated gauge fields
original (full) 50% projected 75% projected 85% projected 90% projected 95% projected center-projected
0.1
λ 0 Im
-0.1
0 0 0 0 0 0 0 Reλ Reλ Reλ Reλ Reλ Reλ Reλ
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 99 / 141 Asqtad Staggered Fermions
original (full) center-projected vortex-removed
0.1 0.1 0.1 λ
Im 0 0 0
-0.1 -0.1 -0.1
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 100 / 141 Asqtad Eigenmode Density Peaks
z=10, t=14 z=11, t=14 z=2, t=7 z=3, t=7
0.0006 0.0006 0.0076 0.0076 0.0005 0.0005 0.0004 0.0004 0.0057 0.0057 0.0003 0.0003 0.0038 0.0038 0.0002 0.0002 0.0001 20 0.0001 20 0.0019 20 0.0019 20 16 16 16 16 0 20 12 0 20 12 0 20 12 0 20 12 16 16 16 16 12 8 y 12 8 y 12 8 y 12 8 y 8 8 8 8 x 4 4 x 4 4 x 4 4 x 4 4
density of eigenvalue #1, maximum 0.000553522856787 at x=5, y=9, z=12, t=14 density of eigenvalue #1, maximum 0.00745768618163 at x=9, y=8, z=4, t=7
0.0006 0.0076 0.0005
0.0057 0.0004
0.0003 0.0038
0.0002 20 0.0019 20 0.0001 16 16
0 20 12 0 20 12 16 16 8 y 8 y 12 12 8 4 8 4 x 4 x 4 0 0 0 0
z=13, t=14 z=14, t=14 z=5, t=7 z=6, t=7
0.0006 0.0006 0.0076 0.0076 0.0005 0.0005 0.0004 0.0004 0.0057 0.0057 0.0003 0.0003 0.0038 0.0038 0.0002 0.0002 0.0001 20 0.0001 20 0.0019 20 0.0019 20 16 16 16 16 0 20 12 0 20 12 0 20 12 0 20 12 16 16 16 16 12 8 y 12 8 y 12 8 y 12 8 y 8 8 8 8 x 4 4 x 4 4 x 4 4 x 4 4
full configuration center-projected
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 101 / 141 Overlap Eigenmode Density Peaks
z=3, t=2 z=4, t=2 z=13, t=3 z=14, t=3
0.0001 0.0001 0.0032 0.0032 8e-05 8e-05 0.0024 0.0024 6e-05 6e-05 0.0016 0.0016 4e-05 4e-05 0.0008 16 0.0008 16 2e-05 16 2e-05 16 12 12 12 12 0 16 0 16 0 16 0 16 12 8 y 12 8 y 12 8 y 12 8 y 8 4 8 4 8 4 8 4 x 4 x 4 x 4 x 4
density of eigenvalue #1, maximum 0.0032086919732 at x=11, y=12, z=5, t=2 density of eigenvalue #1, maximum 8.84376963119e-05 at x=9, y=10, z=15, t=3
0.0001 0.0032 8e-05 0.0024 6e-05
0.0016 4e-05
16 16 0.0008 2e-05 12 12 0 16 0 16 8 8 12 y 12 y 8 4 8 4 4 4 x x 0 0 0 0
z=6, t=2 z=7, t=2 z=0, t=3 z=1, t=3
0.0001 0.0001 0.0032 0.0032 8e-05 8e-05 0.0024 0.0024 6e-05 6e-05 0.0016 0.0016 4e-05 4e-05 0.0008 16 0.0008 16 2e-05 16 2e-05 16 12 12 12 12 0 16 0 16 0 16 0 16 12 8 y 12 8 y 12 8 y 12 8 y 8 4 8 4 8 4 8 4 x 4 x 4 x 4 x 4
full configuration center-projected
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 102 / 141 Correlation between vortices and Dirac modes
Correlator Pi x∈H(V ρλ(x) − hV ρλ(x)i) Cλ = P P Pi x∈H 1 P P
➜ Kovalenko, Morozov, Polikarpov and Zakharov 2005
◮ (vortex) points Pi on the dual lattice ◮ scalar eigenmode density ρλ(x), averaged over the vertices x of the 4d hypercube H, dual to Pi ◮ strongly depends on the number of the vortex plaquettes, attached to a point Pi
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 103 / 141 Vortex correlation for full overlap modes
1.2 zero mode 1st mode 20th mode 1
0.8
0.6
0.4
0.2 vortex correlation
0
-0.2 0 2 4 6 8 10 number of attached vortex plaquettes
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 104 / 141 Vortex correlation for projected overlap modes
0.2 1st mode 20th mode 0.1
0
-0.1
-0.2
-0.3
vortex correlation -0.4
-0.5
-0.6 0 2 4 6 8 10 12 14 number of attached vortex plaquettes
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 105 / 141 Vortex correlation for full asqtad staggered modes
1.5 1st mode 20th mode
1
0.5
vortex correlation 0
-0.5 0 2 4 6 8 10 12 14 number of attached vortex plaquettes
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 106 / 141 Vortex correlation for projected staggered modes
5 1st mode 20th mode
4
3
2
1 vortex correlation
0
-1 0 2 4 6 8 10 12 14 number of attached vortex plaquettes
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 107 / 141 Vortex density at eigenmode peaks
0.18 only overlap eigenmodes overlap and asqtad modes 0.16 only asqtad staggered modes center-projected asqtad staggered
0.14
0.12
0.1 vortex density
0.08
0.06
0.04 0 1 2 3 4 5 6 Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 108 / 141 distance from maximum peak Mechanism of chiral symmetry breaking?
QCD Dynamics breaks Chiral Symmetry
vortex? t quark
Strong indications that Center Vortices explain Chiral Symmetry Breaking.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 109 / 141 Conclusions
◮ Vortices contribute to with fractions to topological charge ◮ Number of zero-modes is related to vortex intersections ◮ Zero mode positions depend on values of Wilson lines ◮ 1 Adjoint zero-modes can’t be attributed to Q = 2 configurations ◮ Even for “admissible” fields indD[A] 6= Q[A] ◮ Dirac monopoles differ from Abelian projected monopoles ◮ Dirac monopoles fading away at large distances have 1 Q = 2 ◮ Strong indications that Center Vortices explain Chiral Symmetry Breaking
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 110 / 141 Phenomena in QCD
Perturbative Phenomena ◮ Asymptotic freedom ◮ Antiscreening Nonperturbative Phenomena ◮ Confinement ◮ Chiral symmetry breaking ◮ Topological Charge and Axial Anomaly ◮ Phase Structure of QCD
Is there a common picture? Center Vortices?
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 111 / 141 Center Vortices 30 years of vortices
➜ ’t Hooft 1979, Nielsen, Ambjorn, Olesen, Cornwall, 1979 Mack, 1980; Feynman, 1981
◮ QCD vacuum is a condensate of closed magnetic flux-lines, they have topology of tubes (3D) or surfaces (4D), ◮ magnetic flux is quantised, it corresponds to the center of the group, quarks feel Aharanov-Bohm effect ◮ Vortex model may explain ... ◮ Confinement → piercing of Wilson loop ≡ crossing of static electric flux tube and moving closed magnetic flux ◮ QCD phase transition: by type of 4D surfaces ◮ Topological charge: vortices carry topological charge at intersection points and writhing points ◮ Spontaneous chiral symmetry breaking ?
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 112 / 141 Lattice QCD action tadpole improved version of the one-loop continuum limit improved SU(2) action of Lüscher and Weisz
➜ Lüscher and Weisz 1985, Poulis 1997
ν ν ν λ µ µ µ standard plaquette, 2 × 1 rectangle and 1 × 1 × 1 parallelogram β β S = β Spl − 2 [1 + 0.2227αs ] Srt − 0.02224 2 αs Spg . 20u0 rt u0 pg Xpl X X 1 ln u u =< Re TrU >1/4, α = −4 0 0 N pl s 1.72597
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 113 / 141 Center Vortices
How to Identify Center Vortices?
➜ Del Debbio, Faber, Greensite, Olejnik (1996–1998)
❏ Fix thermalized SU(2) lattice configurations to maximal center (adj. Landau) gauge by maximizing the expression:
2 A Tr[Uµ(x)] or Tr[Uµ (x)] x,µ x,µ X X +overrelaxation ❏ Make center projection by replacing:
Uµ(x) → Zµ(x) ≡ sign Tr[Uµ(x)]
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 114 / 141 P-Vortex A plaquette is pierced by a P-vortex, if the product of its center projected links gives −1. center vortex in one dimension
center-projected (P-vortex plaquettes)
vortex removed configuration
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 115 / 141 Structure of P-Vortices In 4D they form closed 2D-surfaces in Dual Space, Random Structure
3-dimensional cut through the dual of a 124-lattice.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 116 / 141 Wilson loops A signature for confinement
continuum ··· W (C) = hTr P exp i Aµ(x)dxµ i, IC lattice ··· W (R, T ) = hTrd P Ul i,
l∈C Y Polyakov ··· P(R) = hTrd P U xi. l time C CP ∈C lYP T W (Rˆ, Tˆ )= exp −σˆRˆTˆ − αˆ(Rˆ + Tˆ ) +γ ˆ R space String tension σ W (Rˆ + 1, Tˆ + 1)W (Rˆ, Tˆ ) σ ← χ(Rˆ, Tˆ ) = − ln W (Rˆ, Tˆ − 1)W (Rˆ − 1, Tˆ ) Potential 1 V (R)d = − lim ln W (R, T ) T →∞ T e → C +σ ˆRˆ − Rˆ Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 117 / 141 Confinement potentials
15s 50 24 27 10 15a 6 8 40 3
0 30 (r/a,a)r D V
20
10
0 0.5 1 1.5 2 r/r0
G. Bali, Phys. Rev. D62 (2000) M. Müller et al., Lattice 1991 Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 118 / 141 Area law for center projected loops
−1 −1 −1 −1 Pvortex −1 −1 −1 −1
−1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
denote f the probability that a plaquette has the value -1 hW (A)i = [f (−1) + (1 − f ) · 1]A = exp[ln(1 − 2f ) A], = −σ = exp[−σR × T ], σ ≡− ln| (1{z− 2f }) ≈ 2f Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 119 / 141 Scaling of vortex density
1 two loop L=84 L=124 L=164 L=204
0.1 vortex density
0.01 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 β LW P-vortex surface density vs. coupling constant βLW . “Two loop” line is the scaling prediction with ρ/6Λ2 = 50. q Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 120 / 141 Asymptotic string tension W (R,T ) W (R−1,T −1) Creutz ratios: χ(R, T )= W (R−1,T ) W (R,T −1) → σ 0.35 full theory center-projected 0.3 vortex-removed asymptotic string tension 0.25
) 0.2 R ,
R 0.15 ( χ 0.1
0.05
0
-0.05 0 1 2 3 4 5 6 R Precocious linearity of center projected Creutz ratios. String tension sweeps away the 1/r-potential. Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 121 / 141 P-vortices locate thick center vortices
1.5
1
0.5 0
/ W 0 n W -0.5
-1 W1/W0 W /W -1.5 2 0 0 5 10 15 20 25 Loop Area
n Wn(C)/W0(C) −→ (−1)
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 122 / 141 Thick center vortex model
Vortex crossing the Wilson loop
W (C)= Tr[UU...U] −→ Tr[UU...G...U] Tr G(α ) j=1/2 j=1
G = exp{iαC (x)~nT~ } 1
0 explains Confinement and Screening of Adjoint Sources −1
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 123 / 141 Casimir scaling
2 Fund 6 8 Bali D 10 1.5 15s )) , V e 15mix
(R 24 1
tanh 27
(a Bali
r=2 D 0.5 V
0 2 4 6 8 10 12 14 R Inter quark potential V (R) induced by center vortices, according to the thick vortex model, for quarks in variousx representations.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 124 / 141 Polyakov loops at finite T
0.9 12x2 12x4 0.8 12x6 12x12 0.7
0.6 i
) 0.5 x ~ (
P Z
h 0.4
0.3
0.2
0.1
0 1 1.5 2 2.5 3 3.5 4 4.5 5 βLW
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 125 / 141 Center-projected Polyakov loops at finite T
1 12x2 12x4 0.9 12x6 12x12 0.8
0.7 i ) 0.6 x ~ (
P 0.5 h 2
Z 0.4
0.3
0.2
0.1
0 1 1.5 2 2.5 3 3.5 4 4.5 5 βLW
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 126 / 141 P-Vortices at finite temperature
➜ Engelhardt, Langfeld, Reinhardt, Tennert, 1999
Two 3-dimensional cuts through the dual of a 2 · 123-lattice. Two successive z-slices for the x-y-t-subspace are shown.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 127 / 141 Pontryagin index Q singlet axial current contains anomaly N ∂µj = − f tr(F F˜ ) µ5 16π2 µν µν P-Vortices: closed surfaces of quantised flux
2 ∂x¯µ ∂x¯ν 2 d σµν = ǫab d σ ∂σa ∂σb Q = Topological winding number Q = Self intersection number
➜ Engelhardt, Reinhardt (2000)
1 Q = − ǫ d 2σ d 2σ′ δ4(¯x(σ) − x¯(σ′)) 16 µναβ αβ µν ZS ZS 1 one intersection contributes ± 2 Specify surface orientation !
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 128 / 141 Exact zero-modes and the Atiyah-Singer index theorem
◮ Topological charge:
4 5 Q := d x∂µjµ Z ◮ Axial anomaly: N ∂ j5 = − f tr(F F˜ ) µ µ 16π2 µν µν ◮ Index theorem:
n−, n+: number of left-/right-handed zeromodes
indD[A]= n− − n+ = Q[A]
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 129 / 141 Localisation
◮ Scalar density 2 ρ(x)= |~v(x)cd | , Xc,d c and d, color und Dirac indices
◮ Chiral densities ρ+(x) and ρ−(x)
c,d′ ∗ 1 − γ5 ρ (x)= ~v(x) ~v(x) ′ ± cd 2 cd Xc,d
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 130 / 141 Analytical results
➜ Reinhardt, Schroeder, Tok and Zhukovsky (2002)
Analytical calculations by the Tübingen group. Zero-modes peak at intersections of vortices.
30
20 1
10 0.8
0 0.6 0 0.2 0.4 0.4 0.6 0.2 0.8 1 0
Probability density of zero-mode in the background of four intersecting vortices. Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 131 / 141 Plane Vortices in U(1) subgroup
Geometry Topological charge Fermionic density y y 7.510 7.510 2.5 5 2.5 5 z
10 10
7.5 7.5 z z
5 5
2.5 2.5 y
2.5 5 2.5 5 7.5 7.5 10 10 x x x
Index of the overlap operator: ind D = 0
Topological charge: Q = 0
Conclusion: index theorem valid
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 132 / 141 Conclusions
❏ Confining Disorder ≡ Center Disorder n ❏ P-vortices locate center vortices Wn/W0 = (−1) ❏ Center Dominance: The projected string tension is close to the asymptotic string tension σ χcp(R, R) ≈ σ (R ≥ 2) ❏ Vortex structure changes at the QCD phase transition ❏ Vortices allow for determination of topological charge ❏ Vortex removal restores chiral symmetry ❏ Asqtad staggered fermions show confinement and chiral symmetry breaking also for center-projected configurations ❏ Strong correlations between Dirac eigenmodes and center vortices ❏ Dirac eigenmodes show sharp peaks at intersection and writhing points ❏ Index theorem fulfilled for U(1), but puzzeling for SU(2) vortices
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 133 / 141 Center Dominance
β LW = 2.9
β LW = 3.1
β LW = 3.3 ) 0.1 R,R
( β LW = 3.5 χ
asympt. string tension L=84 L=124 L=164 L=204 0 1 2 3 4 5 6 7 8 R
Center-projected Creutz ratios at βLW = 2.9 − 3.5 obtained after direct Laplacian center gauge fixing. Horizontal bands indicate the asymptotic string tensions. Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 134 / 141 The Overlap Dirac operator
◮ Overlap operator: is of Ginsparg-Wilson type 1 D = 1 + γ ǫ(H+) ov 2 5 L ǫ... sign-functionh , i + HL = γ5DW(−m0)
DW: lattice Wilson Dirac operator for r = 1 for massless fermions: mc < m0 < 2
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 135 / 141 The Staggered Fermion operator
◮ Staggered Fermion operator: is of Ginsparg-Wilson type for fermionfield χ with massless fermions: mc < m0 < 2 1 Dsf = η(x,µ)P(x,µ) 2a µ X † P(x,µ)= U(x,µ)χ(x + aµ) − U (x − aµ,µ)χ(x − aµ) h i x η(x,µ) = (−1) ν(<µ) ν are the staggered fermion phases asqtad formulationP includes next-to-nearest neighbour and staple terms, namely the Naik term, 3-, 5- and 7-staple terms and the Lepage term.
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 136 / 141 Asqtad Staggered Fermions at finite T
123 x 2 123 x 4 123 x 6 124 1.2 1.2 1.2 1.2 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 λ 0 0 0 0 Im -0.2 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 -0.8 -1 -1 -1 -1 -1.2 -1.2 -1.2 -1.2
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 137 / 141 Localization and fractal dimension of eigenmodes The Inverse Participation Ratio (IPR) of a normalized field ρi (x) is defined as N 2 I = N ρi (x). x X=0 Unlocalized: ρ(x)= const. I = 1 δ-function: ρ(x)= δ(x0) I = N localized on fraction f of sites: I = 1/f Scaling properties: a → 0 at fixed volume: I ∼ a4−d L →∞ at fixed a: I ∼ constant. The remaining norm R is defined as n 4 R(n)= 1 − ρi (x) . . . 1 ≤ n ≤ L , i X=1
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 138 / 141 Asqtad IPR of full config vs. β
18 β = 2.9 β = 3.1 β = 3.3 16 β = 3.5 β = 3.7 14
12 i 10 IPR h 8
6
4
2
0 0 2 4 6 8 10 12 14 16 18 20 #λ
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 139 / 141 Asqtad IPR for center-projected config vs. β
1200 β = 2.9 β = 3.1 β = 3.3 β = 3.5 1000 β = 3.7
800 i
600 IPR h
400
200
0 0 2 4 6 8 10 12 14 16 18 20 #λ
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 140 / 141 Asqtad IPR for vortex removed config vs. β
30 β = 2.9 β = 3.1 β = 3.3 β = 3.5 25 β = 3.7
20 i
15 IPR h
10
5
0 0 2 4 6 8 10 12 14 16 18 20 #λ
Manfried Faber (Atominstitut) Vortices and chiral symmetry breaking Feb. 2014 141 / 141