A Study of Lattice Reduction Detection Techniques for LTE ...
Large-N reduction in lattice field theory with adjoint ... A/Wednes… · Large-N reduction in...
Transcript of Large-N reduction in lattice field theory with adjoint ... A/Wednes… · Large-N reduction in...
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
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities 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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities 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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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.
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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.
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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.
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Broad funnelin which volumereduction holds
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Is thisfinite asN→∞?
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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.
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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.
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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.
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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.
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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!
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions
Adjoint Eguchi-Kawai model (AEK) Towards physical quantities Summary & outlook
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!
Mateusz Koren Large-N reduction in lattice field theory with adjoint fermions