Energy-Momentum Tensor and Thermodynamics of Lattice Gauge Theory from Gradient Flow
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Transcript of Energy-Momentum Tensor and Thermodynamics of Lattice Gauge Theory from Gradient Flow
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Energy-Momentum Tensor and Thermodynamics of Lattice Gauge Theory from Gradient Flow
Masakiyo Kitazawa (Osaka U.) 北澤正清(大阪大学)
for FlowQCD CollaborationAsakawa, Hatsuda, Iritani, Itou, MK, Suzuki
FlowQCD, PRD90,011501R (2014)
ATHIC2014, 7 Aug. 2014, Osaka U.
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Lattice QCD
First principle calculation of QCDMonte Carlo for path integral
hadron spectra, chiral symmetry, phase transition, etc.
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Gradient Flow
Luscher, 2010
A powerful tool for various analyses on the lattice
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energymomentum
pressurestress
Poincaresymmetry
Einstein Equation Hydrodynamic Eq.
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: nontrivial observable on the lattice
Definition of the operator is nontrivial because of the explicit breaking of Lorentz symmetry①
② Its measurement is extremely noisydue to high dimensionality and etc.the more severe on the finer lattices
ex:
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因小失大small lose bigstem from
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因小失大miss the wood for the trees
見小利則大事不成孔子(論語、子路 13 )
small lose bigstem from
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If we have
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Thermodynamicsdirect measurement of expectation values
If we have
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Fluctuations andCorrelations
viscosity, specific heat, ...
Thermodynamicsdirect measurement of expectation values
If we have
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Fluctuations andCorrelations
viscosity, specific heat, ...
Thermodynamicsdirect measurement of expectation values
Vacuum Structure
vacuum configuration mixed state on 1st transition
Hadron Structure
confinement string EM distribution in hadrons
If we have
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Themodynamics: Integral Method
: energy density: pressure directly observable
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Themodynamics: Integral Method
: energy density: pressure directly observable
measurements of e-3p for many T vacuum subtraction for each T information on beta function
Boyd+ 1996
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Gradient Flow
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YM Gradient Flow Luscher, 2010
t: “flow time”dim:[length2]
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YM Gradient Flow Luscher, 2010
t: “flow time”dim:[length2]
transform gauge field like diffusion equation
diffusion length
This is NOT the standard cooling/smearingAll composite operators at t>0 are UV finite Luescher,Weisz,2011
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失小得大
coarsegraining
This is NOT the standard cooling/smearing
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Small Flow Time Expansionof Operators and EMT
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Small t Expansion Luescher, Weisz, 2011
remormalized operators
of original theory
an operator at t>0
t0 limit
ori
gin
al 4-d
im t
heory
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Constructing EMTSuzuki, 2013DelDebbio,Patella,Rago,2013
gauge-invariant dimension 4 operators
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Constructing EMT 2
Suzuki coeffs.
Suzuki, 2013
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Constructing EMT 2
Suzuki coeffs.
Remormalized EMT
Suzuki, 2013
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Numerical Analysis: Thermodynamics
Thermodynamicsdirect measurement of expectation values
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Gradient Flow Method
lattice regularizedgauge theory
gradient flow
continuum theory
(with dim. reg.)
continuum theory
(with dim. reg.)gradient flow
analytic(perturbative)
“ 失小得大”
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Gradient Flow Method
lattice regularizedgauge theory
gradient flow
continuum theory
(with dim. reg.)
continuum theory
(with dim. reg.)gradient flow
analytic(perturbative)
measurement on the lattice
“ 失小得大”
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lattice regularizedgauge theory
gradient flow
continuum theory
(with dim. reg.)
continuum theory
(with dim. reg.)gradient flow
Caveats Gauge field has to be sufficiently
smeared!
measurement on the lattice
analytic(perturbative)
Perturbative relation has to be applicable!
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lattice regularizedgauge theory
gradient flow
continuum theory
(with dim. reg.)
continuum theory
(with dim. reg.)gradient flow
Caveats Gauge field has to be sufficiently
smeared!
analytic(perturbative)
Perturbative relation has to be applicable! measurement
on the lattice
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O(t) effect classical
non-perturbativeregion
in continuum
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in continuum
on the lattice
O(t) effect classical
t0 limit with keeping t>>a2
non-perturbativeregion
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Numerical Simulation
• lattice size: 323xNt
• Nt = 6, 8, 10• b = 5.89 – 6.56• ~300 configurations
SU(3) YM theoryWilson gauge action
Simulation 1(arXiv:1312.7492)
SX8 @ RCNPSR16000 @ KEK
using
• lattice size: 643xNt
• Nt = 10, 12, 14, 16• b = 6.4 – 7.4• ~2000 configurations
twice finer lattice!Simulation 2
(new, preliminary)
using BlueGeneQ @ KEKefficiency ~40%
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e-3p at T=1.65Tc
Emergent plateau!
the range of t where the EMT formula is successfully used!
Nt=6,8,10~300 confs.
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e-3p at T=1.65Tc
Emergent plateau!
the range of t where the EMT formula is successfully used!
Nt=6,8,10~300 confs.
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Entropy Density at T=1.65Tc
Emergent plateau!
Direct measurement of e+p on a given T!
systematic
error
Nt=6,8,10~300 confs.
NO integral / NO vacuum subtraction
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Continuum Limit
0
Boyd+1996323xNtNt = 6, 8, 10T/Tc=0.99, 1.24, 1.65
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Continuum Limit
0
Comparison with previous studies
Boyd+1996323xNtNt = 6, 8, 10T/Tc=0.99, 1.24, 1.65
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Numerical Simulation
• lattice size: 323xNt
• Nt = 6, 8, 10• b = 5.89 – 6.56• ~300 configurations
• lattice size: 643xNt
• Nt = 10, 12, 14, 16• b = 6.4 – 7.4• ~2000 configurations
SU(3) YM theoryWilson gauge action
Simulation 1(arXiv:1312.7492)
Simulation 2(new, preliminary)
SX8 @ RCNPSR16000 @ KEK
using using BlueGeneQ @ KEKefficiency ~40%
twice finer lattice!
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Entropy Density on Finer Lattices
The wider plateau on the finer lattices
Plateau may have a nonzero slope
T = 2.31Tc643xNtNt = 10, 12, 14, 162000 confs.
FlowQCD,2013T=1.65Tc
Preliminary!!
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Continuum Extrapolation
0
• T=2.31Tc• 2000 confs• Nt = 10 ~ 16
Nt=16 14 12 10Continuum extrapolationis stable
a0 limit with fixed t/a2
1312.7492
Preliminary!!
Preliminary!!
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Summary
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Summary
EMT formula from gradient flow
This formula can successfully define and calculate the EMT on the lattice. It’s direct
and intuitive
Gradient flow provides us novel methods of various analyses on the lattice avoiding 因
小失大 problem
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Other observablesfull QCD Makino,Suzuki,2014 non-pert. improvement Patella 7E(Thu)
Many Future Studies!!
O(a) improvement Nogradi, 7E(Thu); Sint, 7E(Thu)Monahan, 7E(Thu) and etc.
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Numerical Analysis: EMT Correlators
Fluctuations andCorrelations
viscosity, specific heat, ...
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EMT Correlator
Kubo Formula: T12 correlator shear viscosity
Energy fluctuation specific heat
Hydrodynamics describes long range behavior of Tmn
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EMT Correlator : Noisy…
Nakamura, Sakai, PRL,2005Nt=8improved action~106 configurations … no signal
Nt=16standard action5x104 configurations
With naïve EMT operators
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Energy Correlation Function
T=2.31Tcb=7.2, Nt=162000 confsp=0 correlator
Preliminary!!
smeared
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Energy Correlation Function
T=2.31Tcb=7.2, Nt=162000 confsp=0 correlator
t independent const. energy conservation
Preliminary!!
smeared
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Energy Correlation Function
T=2.31Tcb=7.2, Nt=162000 confsp=0 correlator
specific heatPreliminary!! Novel approach to measure specific heat!smeared
Gavai, Gupta, Mukherjee, 2005
differential method / cont lim.