Large-N pure gauge critical temperature along a line of constant Physics
Large-N pure gauge critical temperature along aline of constant Physics
Jamie Hudspith Anthony Francis
York University, Toronto
April 21, 2017
Large-N pure gauge critical temperature along a line of constant Physics
Introduction
Plan of the talk
1. BackgroundI DefinitionsI Lattice geometry
2. Scale SettingI Gradient flow for Large NI Planar limit
3. Finite-T simulationsI Determining the critical βI
√t0TC
4. Large-N limit
Large-N pure gauge critical temperature along a line of constant Physics
Introduction
Why?
Theoretical and Technical.
I Various BSM theories rely on SU(N) with some N
I Large-N limit is used in many approaches, simpler than QCD andsome analytic calculations can be performed
I Confinement is a defining feature of SU(N) gauge theories
I No fermions so simulations are cheap, although inevitably thecomputation cost increases
I Some commonly-used techniques are interesting to extend to largematrices
I No studies for a few years, probably worth an update especially withsome different methods
Large-N pure gauge critical temperature along a line of constant Physics
Introduction
Of course I am not the only one to think about this
Lucini, Teper and Wenger (2005)”Properties of the deconfining phase transition in SU(N) gauge theories”
Lucini, Rago and Rinaldi (2012)”SU(Nc) gauge theories at deconfinement”
Nb: Lucini et al use the string tension σ to set the scale and thePolyakov loop susceptibility to measure βC , extrapolating βC to infinitevolume using small volumes.
Francis, A. and Kaczmarek, O. and Laine, M. and Neuhaus, T. andOhno, H. ”Critical point and scale setting in SU(3) plasma: An update”Nb:SU(3) only
Use more or less the same method for pure gauge SU(N).
Large-N pure gauge critical temperature along a line of constant Physics
Introduction
Notation
Uµ
(x + a
µ
2
)= e iag0Aµ(x+a µ
2 ). (1)
We use the Wilson Plaquette action, V = L3s × Lt
Sg = Vβ (N − Up) = a4E 2 + ..
Up =1
V
∑x,µ<ν
<(
Tr
[Uµν
(x + a
µ
2+ a
ν
2
)]),
E 2 =∑x,µ,ν
Fµν(x)Fµν(x).
(2)
l =1
NL3
L3∑x
Tr
[Lt∏t=1
Ut
(x + a
t
2
)]. (3)
Large-N pure gauge critical temperature along a line of constant Physics
Introduction
Deconfinement
I Ls/a ≤ Lt/a = ”Zero Temperature”
I Lt/a < Ls/a = ”Finite Temperature”
Inverse anisotropic length is temperature,
aT =a
Lt(4)
All pure-gauge SU(N) theories show confinement, with an expectedfirst order phase transition for N > 2.
Polyakov loop is the order parameter of this transition,
|l | ≈ e−F . (5)
|l | = 0, F =∞ |l | 6= 0,F = finite.Breaking of center symmetry l ∈ ZN .
Large-N pure gauge critical temperature along a line of constant Physics
Introduction
SU(N) starfish
Large-N pure gauge critical temperature along a line of constant Physics
Introduction
Geometry
Even-dimension checker-boarding
Checker-board updating alternating R and B.B R B R
R B R B
B R B R
R B R B
,
Large-N pure gauge critical temperature along a line of constant Physics
Introduction
Geometry
Odd-dimension checker-boarding
Checker-board updating alternating R and B and G.G R B
B G R
R B G
Sometimes NG < NG ,NB but we loop these in a multi-threaded code so
it doesn’t really matter.
https://github.com/RJHudspith/GLU
Large-N pure gauge critical temperature along a line of constant Physics
Scale Setting
Scale setting overview
Traditionally lattice spacing a (in QCD) is determined by matching toexperimental quantities amΩ, amK , amπ, afπ ... etc.
There aren’t any for large N, but can match to auxiliary scales such as:
I The string tension a√σ
I The Sommer parameter ar0
I The Gradient Flow scale a√t0
Large-N pure gauge critical temperature along a line of constant Physics
Scale Setting
Gradient Flow
Solve,U = Z (U)U, U(t + ε) = e iεZ(U(t))U(t). (6)
su(N) matrix exponential! should be done by a well-convergent series,trivially,
(eA/(2)n)2n
= eA. (7)
Using a (4,4) pade representation of the exponential,
eA =I + A(C0 + A(C1 + A(C2 + AC3)))
I − A(C0 − A(C1 − A(C2 − AC3))),
C0 =1
2, C1 =
3
28, C2 =
1
84, C3 =
1
1680,
(8)
and n = 3 and 4 to perform this exponential and final reunitarisation stepto remain in the group.
Large-N pure gauge critical temperature along a line of constant Physics
Scale Setting
Walking on a line of constant PhysicsUsually we introduce a reference scale t0/a2 as,
t2E 2|t=t0 = CE . (9)
The beta function diverges in the large-N limit, we must take the planarlimit Λ = Ng 2. But wait! E 2 ∝ g 2
0 so we can define a planar limit scalewhich I will confusingly call t0/a2,
t2E 2|t=t0 = CEN. (10)
Where we will choose CE = 0.1 so that we can compare our SU(3) resultto the literature.Scale set at zero temperature, can form a quantity with constant cut offeffects
L√t0
= CL. (11)
We will tune our ensembles to CL = 10.,
Large-N pure gauge critical temperature along a line of constant Physics
Scale Setting
Tuning for SU(3), SU(4) and SU(5)
5.7 5.75 5.8 5.85 5.9 5.95 6 6.05 6.1
9.9
10
10.1 SU( 3 )
10.5 10.6 10.7 10.8 10.9 11 11.1
9.9
10
10.1 SU( 4 )
16.7 16.8 16.9 17 17.1 17.2 17.3 17.4 17.5 17.6
β
9.9
10
10.1 SU( 5 )
CL =
Ls /
( t
0 )
(1/2
)
Large-N pure gauge critical temperature along a line of constant Physics
Scale Setting
Tuning for SU(6), SU(7) and SU(8)
24.3 24.4 24.5 24.6 24.7 24.8 24.9 25 25.1 25.2 25.3 25.4 25.5
9.9
10
10.1 SU( 6 )
33.2 33.4 33.6 33.8 34 34.2 34.4 34.6 34.8
9.9
10
10.1 SU( 7 )
43.5 43.75 44 44.25 44.5 44.75 45 45.25
β
9.9
10
10.1 SU( 8 )
CL =
L /
( t
0 )
(1/2
)
Large-N pure gauge critical temperature along a line of constant Physics
Scale Setting
√t0 as a function of β
0 0.25 0.5 0.75 1 1.25 1.5 1.75
β − βL=10
1
1.25
1.5
1.75
2
2.25(
t 0 )
(1/2
) / a
SU( 3 )
SU( 4 )
SU( 5 )
SU( 6 )
SU( 7 )
SU( 8 )
Large-N pure gauge critical temperature along a line of constant Physics
Finite Temperature
More constants
We define the critical coupling βC as the point where deconfinementoccurs
a(βC )TC =a(βC )
Lt. (12)
We know how our auxiliary scale behaves with β so we can define yetanother dimensionless quantity,
√t0TC =
√t0
Lt= CT . (13)
This is, in principle, a physical quantity that other theorists can use onceit is known in the continuum.
Large-N pure gauge critical temperature along a line of constant Physics
Finite Temperature
The separatrix
Originally considered as a method of determining the phase transition ofthe n-state Potts model with exponentially-small finite volumecorrections.
From SU(3) we see that anisotropy Ls
Lt= CA = 3 is very close to the
infinite volume limit. So we choose to work solely at CA = 3.
With a double-peaked histogram of |l | with W measurements. We countthe number of points below the minimum between the two maxima, thisquantity we call wg ,
S(β) =(q + 1)wg −W
(q − 1)wg + W. (14)
S(β) = 1 confined, S(β) = −1 deconfined.q is the q-fold degeneracy weight, which we set to NC .
Large-N pure gauge critical temperature along a line of constant Physics
Finite Temperature
Separatrix: The Movie
Large-N pure gauge critical temperature along a line of constant Physics
Finite Temperature
SU(3) has small finite volume corrections
Figure: Exponentially-reduced finite volume effects in determining the criticalcoupling as compared to the Polyakov loop susceptibility method. Plot takenfrom Francis-2014.
Large-N pure gauge critical temperature along a line of constant Physics
Finite Temperature
Comparisons 1
Lt/a Francis-2014 Us
SU(3)
5 5.8000(5)† 5.8003(2)6 5.8943(3) 5.8954(5)7 - 5.9810(4)8 6.0624(4) 6.0624(3)
Lt/a Lucini-2005 Us
SU(4)
5 10.6373(5) 10.6398(1)6 10.7898(16) 10.7955(3)7 10.9415(12) 10.9460(6)8 10.0880(22) 11.0880(3)
Table: Comparison of our results with those in the literature for βC for thegauge groups SU(3) and SU(4). The result of † is from Lucini et al 2004hep-lat/0307017.
Large-N pure gauge critical temperature along a line of constant Physics
Finite Temperature
Comparisons 2
Lt/a Lucini-2012 Us
SU(5)
5 16.8762(12) 16.8764(1)6 17.1074(33) 17.1105(2)7 17.3386(31) 17.3294(5)8 17.5585(36) 17.5516(6)
I SU(6), SU(7) and SU(8) data is still being gathered and analyzed.
I Transition becomes more first order as N and Volume increase,requires a large number of ensembles.
Large-N pure gauge critical temperature along a line of constant Physics
Finite Temperature
Continuum extrapolations of√t0TC
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
a2 / t
0
0.235
0.24
0.245
0.25
0.255
0.26(
t 0 )
(1/2
) TC
SU( 3 )
SU( 4 )
SU( 5 )
Large-N pure gauge critical temperature along a line of constant Physics
Finite Temperature
Large-N limit?
0 0.02 0.04 0.06 0.08 0.1 0.12
1 / N2
0.235
0.2375
0.24
0.2425
0.245
0.2475
0.25
0.2525
0.255(
t 0 )
(1/2
) TC
Large-N pure gauge critical temperature along a line of constant Physics
Conclusions
Conclusions
I We have accurately set the scale and defined a line of constantPhysics that we can use for large-N extrapolations.
I We have utilised the Separatrix to determine the critical β values forN ≥ 3 and found them more or less in accordance with theliterature.
I We have demonstrated you can accurately obtain the dimensionlessquantity
√t0TC and that the approach to the large-N limit is fairly
mild.