Large-N reduction in lattice field theory with adjoint ... A/Wednes… · Large-N reduction in lattice eld theory with adjoint fermions Mateusz Koren Jagiellonian University, Cracow

Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook

Large-N reduction in lattice field theory withadjoint fermions

Mateusz Koren

Jagiellonian University, Cracow

In collaboration with Barak Bringoltz and Stephen R. Sharpe

Phys. Rev. D 85, 094504 (2012) + more

Project carried within the MPD programme ”Physics of Complex Systems” of the Foundation for Polish Science andco-financed by the European Regional Development Fund in the framework of the Innovative Economy Programme.

Lattice 2012, Cairns, 27-06-2012

Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions


Table of contents

1 Adjoint Eguchi-Kawai model (AEK)Volume reductionDefinition of the modelPhase diagram of AEK

2 Towards physical quantitiesWilson loopsMeson masses

3 Summary & outlook



Volume reduction

Eguchi, Kawai, 82: Large volume lattice YM theory ≡ singlesite YM when N →∞ if center symmetry (ZN)4 unbroken

Bhanot, Heller, Neuberger, 82: Z4N broken, as seen in PT

Several fixes, none of them fully satisfactory

Kovtun, Unsal, Yaffe, 04: Modern view of reduction (large-Norbifold equivalence):

SU(N) gaugetheory on Ld

SU(N') gauge theoryon 1 , N' = L Nd d

Restrict to 0-momentum fields

Use gauge & center transformationsto map blocks of SU(N')to different lattice points

KUY, 07: Add massless adjoint fermions with PBC ⇒reduction holds in PT



Volume reduction

Eguchi, Kawai, 82: Large volume lattice YM theory ≡ singlesite YM when N →∞ if center symmetry (ZN)4 unbroken

Bhanot, Heller, Neuberger, 82: Z4N broken, as seen in PT

Several fixes, none of them fully satisfactory

Kovtun, Unsal, Yaffe, 04: Modern view of reduction (large-Norbifold equivalence):

SU(N) gaugetheory on Ld

SU(N') gauge theoryon 1 , N' = L Nd d

Restrict to 0-momentum fields

Use gauge & center transformationsto map blocks of SU(N')to different lattice points

KUY, 07: Add massless adjoint fermions with PBC ⇒reduction holds in PT



Definition of AEK model

Adjoint Eguchi-Kawai (EK + adj. Wilson fermions):

Z =∫

D[U, ψ, ψ] e(Sgauge+PNf

j=1 ψj DW ψj )

Sgauge = 2Nb∑

µ<ν ReTrUµUνU†µU†ν , b = 1

g2N

DW = 1− κ[∑4

µ=1 (1− γµ) Uadjµ + (1 + γµ) U†adj

µ

]

Bringoltz, Sharpe, 09: For Nf = 1 even heavy fermionsstabilize center symmetry!

Azeyanagi, Hanada, Unsal, Yacoby, 10: semi-analyticunderstanding of this result

Probes of Z4N : general “open loops”:

Kn ≡ 1N TrUn1

1 Un22 Un3

3 Un44 , with nµ = 0,±1,±2, . . .

In particular, we find Polyakov loops TrUµ and “cornervariables” TrUµU±1

ν very helpful.





Z =∫


j=1 ψj DW ψj )

Sgauge = 2Nb∑


g2N

DW = 1− κ[∑4


µ

]Bringoltz, Sharpe, 09: For Nf = 1 even heavy fermionsstabilize center symmetry!



Kn ≡ 1N TrUn1

1 Un22 Un3

3 Un44 , with nµ = 0,±1,±2, . . .


ν very helpful.





Z =∫


j=1 ψj DW ψj )

Sgauge = 2Nb∑


g2N

DW = 1− κ[∑4


µ

]Bringoltz, Sharpe, 09: For Nf = 1 even heavy fermionsstabilize center symmetry!



Kn ≡ 1N TrUn1

1 Un22 Un3

3 Un44 , with nµ = 0,±1,±2, . . .


ν very helpful.



Phase diagram of Nf = 2 AEK

ZN4

ZN4

ZN4 Z1Z2Z5 Z3

b

κc

1st order

regionHysteresis

crossover or bulk

Z Z1 2

κ

0.04 0.08 0.16 0.20 0.240.120

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Z4

transition

κf




ZN4

ZN4

ZN4 Z1Z2Z5 Z3

b

κc

1st order

regionHysteresis

crossover or bulk

Z Z1 2

κ

0.04 0.08 0.16 0.20 0.240.120

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Z4

transition

κf

Broad funnelin which volumereduction holds




ZN4

ZN4

ZN4

Z1Z2Z5 Z3

b

κc

1st order

regionHysteresis

crossover or bulk

Z Z1 2

κ

0.04 0.08 0.16 0.20 0.240.120

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Z4

transition

κf

Is thisfinite asN→∞?



Phase diagram cntd. Funnel edge (κf ) vs N

Evidence of finite width of the funnel at b = 1 as N →∞

0.04

0.042

0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

0.06

0.015 0.02 0.025 0.03 0.035 0.04 0.045

κ

1/N

κf , b=1.0

Fit: 0.51(2)/N + 0.0655(5), χ2/d.o.f.=0.039

Fit: 64(9)/N2- 4.5(3)/N + 1/8, χ2/d.o.f.=23



Phase diagram cntd. Funnel edge (κf ) vs N

Evidence of finite width of the funnel at b = 1 as N →∞

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.015 0.02 0.025 0.03 0.035 0.04 0.045

κ

1/N

κf , Nf=1, b=1.0

κf , Nf=2, b=1.0

Fit: 0.747(11)/N + 0.0937(3), χ2/d.o.f.=0.014

Fit: 35(5)/N2- 2.91(15)/N + 1/8, χ2/d.o.f.=2.9



Wilson loops

For given µ, ν: W (M, L) = 〈 1N TrUM

µ ULνU†Mµ U†Lν 〉 ∼ e−V (M)L

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0 10 20 30 40 50 60

log(

W)

L

N=10N=21N=37N=47N=53

Log of 1× L Wilson loops vs L for various N



Wilson loops cntd.

Rise of W (M, L) at large L – finite N corrections

It can be seen in strong coupling expansion – contrary to largevolume one never encounters integrals

∫dUUij = 0 – zero-th

order does not disappear.

In pure gauge (for 0 < L,M ≤ N):

W (L,M)b,κ→0−→ 1

N2 − 1

(L + M − 1− LM/N2

)

Idea: use 24 lattices with only odd values of L,M.



Wilson loops cntd.

Rise of W (M, L) at large L – finite N corrections

It can be seen in strong coupling expansion – contrary to largevolume one never encounters integrals

∫dUUij = 0 – zero-th

order does not disappear.

In pure gauge (for 0 < L,M ≤ N):

W (L,M)b,κ→0−→ 1

N2 − 1

(L + M − 1− LM/N2

)Idea: use 24 lattices with only odd values of L,M.



Wilson loops cntd.

-6

-5

-4

-3

-2

-1

0

0 2 4 6 8 10 12

log(

W)

L

W(1,L)

W(2,L)

W(3,L)

W(4,L)

Wilson loops on 24 lattice, N = 10, b = 0.35, κ = 0.1



Wilson loops cntd.

-6

-5

-4

-3

-2

-1

0

0 2 4 6 8 10 12

log(

W)

L

W(1,L)

W(3,L)

Wilson loops on 24 lattice, N = 10, b = 0.35, κ = 0.1



Meson masses

Narayanan, Neuberger, 05: “quenched momentumprescription”

Multiply temporal links by additional U(1) factors interpretedas “quenched momentum”: U4 → U4 e ip

Mπ(p,m) = Tr{γ5 D−1(U4 e ip/2,m) γ5 D−1(U4 e−ip/2,m)

},

First results: hard to obtain light meson masses. Needlarger N / further improvements.

Caveat: While legitimate for fundamental mesons, can this beextended to adjoint mesons?

Future plan: Compare with one extended dimension.



Meson masses

Narayanan, Neuberger, 05: “quenched momentumprescription”

Multiply temporal links by additional U(1) factors interpretedas “quenched momentum”: U4 → U4 e ip

Mπ(p,m) = Tr{γ5 D−1(U4 e ip/2,m) γ5 D−1(U4 e−ip/2,m)

},

First results: hard to obtain light meson masses. Needlarger N / further improvements.

Caveat: While legitimate for fundamental mesons, can this beextended to adjoint mesons?

Future plan: Compare with one extended dimension.



Summary & outlook

Summary:

Center symmetry is unbroken in AEK in a wide range ofphysically interesting parameters ⇒ volume reduction works!

“No free lunch” – calculation of physical quantities requireslarge values of N.

TODO list:

Make 1/N corrections smaller – use slightly larger lattices(+parallelism), twisting, better actions?

Perform simulations with one “long” dimension – meson &glueball masses, thermodynamics. . .

Thank you for your attention!



Summary & outlook

Summary:

Center symmetry is unbroken in AEK in a wide range ofphysically interesting parameters ⇒ volume reduction works!

“No free lunch” – calculation of physical quantities requireslarge values of N.

TODO list:

Make 1/N corrections smaller – use slightly larger lattices(+parallelism), twisting, better actions?

Perform simulations with one “long” dimension – meson &glueball masses, thermodynamics. . .

Thank you for your attention!


Large-N reduction in lattice field theory with adjoint ... A/Wednes… · Large-N reduction in lattice eld theory with adjoint fermions Mateusz Koren Jagiellonian University, Cracow

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