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Aug 18, 2023 132 likes •446 views

Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD Marc Klegrewe and Wolfgang Unger Bielefeld University Lattice 2018 East Lansing, 27th July 2018 Marc Klegrewe Temporal Correlators in the Continuous Time Limit

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Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD

Marc Klegrewe and Wolfgang Unger Bielefeld University

Lattice 2018

East Lansing, 27th July 2018

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 1 / 14

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QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit

Study regime where sign problems can be made mild: ⇒ limit of infinite gauge coupling g → ∞, β = 2Nc g2 → 0

Problem

[Wolff & Rossi, 1984] Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 2 / 14

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QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit

Study regime where sign problems can be made mild: ⇒ limit of infinite gauge coupling g → ∞, β = 2Nc g2 → 0

Problem

[Wolff & Rossi, 1984]

A change of integration order results in SC-partition function for staggered fermions: ZSC =

  • {n,k,l}
  • x

Nc! nx! (2amq)nx

  • monomers
  • b=(x,µ)

(Nc − kb)! Nc!kb! γ2kbδµ0

  • mesonic hoppings/dimers
  • l

w(l, µ)

  • baryonic hoppings

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 2 / 14

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QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit QCD in the Strong Coupling Limit

Study regime where sign problems can be made mild: ⇒ limit of infinite gauge coupling g → ∞, β = 2Nc g2 → 0

Problem

[Wolff & Rossi, 1984]

A change of integration order results in SC-partition function for staggered fermions: ZSC =

  • {n,k,l}
  • x

Nc! nx! (2amq)nx

  • monomers
  • b=(x,µ)

(Nc − kb)! Nc!kb! γ2kbδµ0

  • mesonic hoppings/dimers
  • l

w(l, µ)

  • baryonic hoppings

Fully combinatorial problem, restricted by Grassmann constraint: nx +

  • ±µ

kxµ = Nc,

  • ±µ

lxµ = 0, ∀x In the following restrict to chiral limit where monomer density n = 0

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 2 / 14

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Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD

First: Introduction of anisotropy for continuous temperature variation: aT = 1 Nτ ⇒ aT = ξ(γ) Nτ , ξ(γ) = a/aτ

  • anisotropy parameter

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 3 / 14

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Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD

First: Introduction of anisotropy for continuous temperature variation: aT = 1 Nτ ⇒ aT = ξ(γ) Nτ , ξ(γ) = a/aτ

  • anisotropy parameter

Second: Gamma dependence of ξ(γ) non trivial:

[de Forcrand, Unger & Vairinhos, 2018]

ξ(γ) ≈ κγ2 + γ2 1 + λγ4 , κ = 0.781 for SU(3)

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 3 / 14

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Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD

First: Introduction of anisotropy for continuous temperature variation: aT = 1 Nτ ⇒ aT = ξ(γ) Nτ , ξ(γ) = a/aτ

  • anisotropy parameter

Second: Gamma dependence of ξ(γ) non trivial:

[de Forcrand, Unger & Vairinhos, 2018]

ξ(γ) ≈ κγ2 + γ2 1 + λγ4 , κ = 0.781 for SU(3) Definition of the Continuous Time Limit as: Nτ → ∞, γ → ∞ with ξ(γ) Nτ = κγ2 Nτ = aT fixed

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 3 / 14

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Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD Continuous Time limit within Strong Coupling QCD

First: Introduction of anisotropy for continuous temperature variation: aT = 1 Nτ ⇒ aT = ξ(γ) Nτ , ξ(γ) = a/aτ

  • anisotropy parameter

Second: Gamma dependence of ξ(γ) non trivial:

[de Forcrand, Unger & Vairinhos, 2018]

ξ(γ) ≈ κγ2 + γ2 1 + λγ4 , κ = 0.781 for SU(3) Definition of the Continuous Time Limit as: Nτ → ∞, γ → ∞ with ξ(γ) Nτ = κγ2 Nτ = aT fixed Continuous Time partition function: ZCT(T) =

  • k∈2N

1

2aT

G′∈Γk

eµBB/T ˆ νNT

T

with k =

  • b=(x,ˆ

i)

kb, NT =

  • x

nT (x)

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 3 / 14

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Benefits and Comments on Continuous Time Limit Benefits and Comments on Continuous Time Limit Benefits and Comments on Continuous Time Limit Benefits and Comments on Continuous Time Limit Benefits and Comments on Continuous Time Limit

  • No discretization errors due to finite Nτ
  • Only one parameter left (temperature T)
  • Baryons become static for Nc ≥ 3 ⇒ no extend in spatial direction

⇒ Sign problem is absent

  • Baryons are massive (even though in chiral limit)
  • No multiple spatial dimers (suppressed by γ)
  • Faster algorithm for medium to large temporal extends

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 4 / 14

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Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm

  • Worm-type Monte Carlo Algorithm

[Adams & Chandrasekharan, 2003]

  • absorption (even) and emission (odd) site decomposition of lattice

Mesonic worm update:

  • Place tail of mesonic worm on lattice at absorption site

⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site

t

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14

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Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm

  • Worm-type Monte Carlo Algorithm

[Adams & Chandrasekharan, 2003]

  • absorption (even) and emission (odd) site decomposition of lattice

Mesonic worm update:

  • Place tail of mesonic worm on lattice at absorption site

⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site

T H t

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14

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Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm

  • Worm-type Monte Carlo Algorithm

[Adams & Chandrasekharan, 2003]

  • absorption (even) and emission (odd) site decomposition of lattice

Mesonic worm update:

  • Place tail of mesonic worm on lattice at absorption site

⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site

T H t

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14

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Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm

  • Worm-type Monte Carlo Algorithm

[Adams & Chandrasekharan, 2003]

  • absorption (even) and emission (odd) site decomposition of lattice

Mesonic worm update:

  • Place tail of mesonic worm on lattice at absorption site

⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site

T H t

t

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14

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SLIDE 14

Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm

  • Worm-type Monte Carlo Algorithm

[Adams & Chandrasekharan, 2003]

  • absorption (even) and emission (odd) site decomposition of lattice

Mesonic worm update:

  • Place tail of mesonic worm on lattice at absorption site

⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site

T H t

a e ?

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14

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Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm

  • Worm-type Monte Carlo Algorithm

[Adams & Chandrasekharan, 2003]

  • absorption (even) and emission (odd) site decomposition of lattice

Mesonic worm update:

  • Place tail of mesonic worm on lattice at absorption site

⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site

T H t

a

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14

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SLIDE 16

Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm

  • Worm-type Monte Carlo Algorithm

[Adams & Chandrasekharan, 2003]

  • absorption (even) and emission (odd) site decomposition of lattice

Mesonic worm update:

  • Place tail of mesonic worm on lattice at absorption site

⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site

T H t

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14

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SLIDE 17

Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm Continuous Time Algorithm

  • Worm-type Monte Carlo Algorithm

[Adams & Chandrasekharan, 2003]

  • absorption (even) and emission (odd) site decomposition of lattice

Mesonic worm update:

  • Place tail of mesonic worm on lattice at absorption site

⇒ Violation of Grassmann constraint ⇒ propagate head restoration of Grassmann constraint if head at emission site Weight of configuration ruled by spatial dimer emission/absorption

  • spatial dimer emission ruled by Poisson process
  • Vertex weights decide spatial dimer absorption

P(∆β) ∼ exp(λ∆β), ∆β ∈ [0, β = 1/aT] "decay constant" λ for spatial dimer emission: λ = dD(x)/4, dM(x) = 2d −

µ nB(x ± ˆ

µ) with dM(x) the number of mesonic neighbors

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 5 / 14

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Two-point Correlators Two-point Correlators Two-point Correlators Two-point Correlators Two-point Correlators

  • Sample monomer-monomer two-point correlation functions

C(tH − tT, xH − xT) = C(τ, x)

  • accumulate observables during worm evolution

(tail absorption/source, head emission/sink) C(τ, x) = Nc O(C(τ, x)) #worm updates

  • Measure Chiral Susceptibility χσ by summing over worm estimators:

χσ = 1 V

  • x

C(τ, x) Discrete Time: O(C(τ, x)) → O(C(τ, x)) + f (γ) · δxT ,x1δxH,x2, τ ∈ [0, . . . Nτ] Continuous Time: O(C(τ, x)) → O(C(τ, x)) + g(T) · δxT ,x1δxH,x2, τ ∈ [0, . . . 1

T ]

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 6 / 14

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Extracting Meson Masses Extracting Meson Masses Extracting Meson Masses Extracting Meson Masses Extracting Meson Masses

  • Extract pole masses for temporal correlators with zero spatial momentum

E0( p = 0) = m0, C(τ) =

  • x

¯ χ0χ0 ¯ χ

x,tχ x,tgD x

  • For staggered fermions: Restrict to diagonal of Dirac-taste-kernel (Nf = 1)

gD

x

ΓD ⊗ ΓF JPC Physical states NO O NO O NO O 1 1 ⊗ 1 γ0γ5 ⊗ (γ0γ5)∗ 0++ 0−+ σS πA (−1)xi γiγ5 ⊗ (γiγ5)∗ γiγ0 ⊗ (γiγ0)∗ 1++ 1−− aA ρT (−1)xj +xk γjγk ⊗ (γjγk)∗ γi ⊗ γ∗

i

1+− 1−− bT tρV (−1)xi +xj +xk γ0 ⊗ γ∗ γ5 ⊗ (γ5)∗ 0+− 0−+ −V tπPSt tt channel of primary interest

[Altmeyer et al., 1993] Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 7 / 14

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Temporal Correlators in Continuous Time Temporal Correlators in Continuous Time Temporal Correlators in Continuous Time Temporal Correlators in Continuous Time Temporal Correlators in Continuous Time

  • Introduce binning
  • Evaluate at same spatial site ⇒ Zero momentum projection
  • Accumulate histograms while worm head propagates

Value per bin:

g(T) #bins

  • Distinguish histograms for even and odd interval contributions

Extract pole masses from temporal correlators + Study various channels In CT: Masses measured in units of M/T

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 8 / 14

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Discrete and Continuous Time Correlator Fits Discrete and Continuous Time Correlator Fits Discrete and Continuous Time Correlator Fits Discrete and Continuous Time Correlator Fits Discrete and Continuous Time Correlator Fits

Discrete time: 4 parameter fit Either combined: C(τ) = aNO cosh(mNO(τ − Nτ/2)) − aO cos(πτ) cosh(mO(τ − Nτ/2)) Or split up for Even and Odd histograms: CDT,Even(τ) = aNO cosh(mNO(τ − Nτ/2))−aO cosh(mO(τ − Nτ/2) CDT,Odd(τ) = aNO cosh(mNO(τ − Nτ/2))

  • Non-oscillating Correlator

+ aO cosh(mO(τ − Nτ/2)

  • Oscillating Correlator

⇒ CDT,NO(τ) = 1 2 (CDT,Even(τ) + CDT,Odd(τ)) , CDT,O(τ) = 1 2 (CDT,Even(τ) − CDT,Odd(τ))

81.4 81.6 81.8 82 82.2 82.4 82.6 2 4 6 8 Nτ Nτ=16 C(τ) CDT,Odd(τ) CDT,Even(τ) CDT,NO(τ) Even/Odd histogram data

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 9 / 14

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Discrete and Continuous Time Correlator Fits Discrete and Continuous Time Correlator Fits Discrete and Continuous Time Correlator Fits Discrete and Continuous Time Correlator Fits Discrete and Continuous Time Correlator Fits

Discrete time: 4 parameter fit Either combined: C(τ) = aNO cosh(mNO(τ − Nτ/2)) − aO cos(πτ) cosh(mO(τ − Nτ/2)) Or split up for Even and Odd histograms: CDT,Even(τ) = aNO cosh(mNO(τ − Nτ/2))−aO cosh(mO(τ − Nτ/2) CDT,Odd(τ) = aNO cosh(mNO(τ − Nτ/2))

  • Non-oscillating Correlator

+ aO cosh(mO(τ − Nτ/2)

  • Oscillating Correlator

⇒ CDT,NO(τ) = 1 2 (CDT,Even(τ) + CDT,Odd(τ)) , CDT,O(τ) = 1 2 (CDT,Even(τ) − CDT,Odd(τ)) Continuous time: 2/4 parameter fit of added and subtracted histograms respectively CCT,NO(τ) = aNO cosh(mNO(τ − 1/2)) =1 2(COdd (τ) + CEven(τ)) CCT,O(τ) = aO cosh(mO(τ − 1/2)) =1 2(COdd(τ) − CEven(τ))

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 9 / 14

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From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses

166 168 170 172 174 176 178 180 5 10 15 20 25 30 C(τ) Nτ 16 20 24 28 32 36 40 44 48 52 56 60

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 10 / 14

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From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.02 0.04 0.06 aτM 1/Nτ πPS 0.56 0.565 0.57 0.575 0.58 0.585 0.59 0.595 0.6 0.605 0.61 0.02 0.04 0.06 M/T 1/Nτ πPS CT Change aτM → M/T in order to compare with Continuous Time results

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 10 / 14

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From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses

1 2 3 4 5 6 7 8 9 0.6 0.8 1 1.2 1.4 1.6 1.8 2 M/T aT σS aA bT −V πA ρT ρV πPS

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 10 / 14

Preliminary

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From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses From discrete Histograms to Correlators and Masses

0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 aM aT σS aA bT −V πA ρT ρV πPS

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 10 / 14

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Continuous Time Correlators Continuous Time Correlators Continuous Time Correlators Continuous Time Correlators Continuous Time Correlators

10−07 10−06 10−05 10−04 10−03 10−02 0.1 0.2 0.3 0.4 0.5 C(τ) Lτ πPS T=0.500 T=1.000 T=1.500 T=2.000 10−07 10−06 10−05 10−04 10−03 10−02 0.1 0.2 0.3 0.4 0.5 C(τ) Lτ ρV T=0.500 T=1.000 T=1.500 T=2.000

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 11 / 14

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SLIDE 28

Four Parameter Fit Four Parameter Fit Four Parameter Fit Four Parameter Fit Four Parameter Fit

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

aT

10−1 100 101

M/T

CT DT σS / πPS πA / −V bT(γ2γ3) / ρV(γ2) ρV(γ1) / bT(γ1γ3) σS / πPS πA / −V bT(γ2γ3) / ρV(γ2) ρV(γ1) / bT(γ1γ3) Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 12 / 14

Preliminary

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Behaviour around chiral transition Behaviour around chiral transition Behaviour around chiral transition Behaviour around chiral transition Behaviour around chiral transition

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

aT

0.2 0.3 0.4 0.5

M/T

σS / πPS 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

aT

5 10 15 20

M/T

ρV(γ1) / bT(γ1γ3) Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 13 / 14

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Behaviour around chiral transition Behaviour around chiral transition Behaviour around chiral transition Behaviour around chiral transition Behaviour around chiral transition

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

aT

0.0 0.2 0.4 0.6 0.8 1.0

aM

σS / πPS 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

aT

2 4 6 8 10 12

aM

ρV(γ1) / bT(γ1γ3) Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 13 / 14

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Summary and Outlook Summary and Outlook Summary and Outlook Summary and Outlook Summary and Outlook

  • Measured monomer-monomer two-point functions → constructed temporal

correlators

  • On our way to extract and compare pole masses for discrete and continuous time
  • Consider excited states, especially for low temperatures ⇒ Mass extraction and

analysis not yet fully completed

  • Obtain diffusion constant from zero momentum meson correlators

⇒ Extract spectral function from correlation data C(τ, T) =

∞

dωK(ω, τ)σ(ω, T) =

∞

dω cosh(τ(ω −

1 2T ))

sinh( ω

2T )

σ(ω, T) Typical bottleneck: #data points in temporal direction → advantage of large binning

  • Reconstruct spectral function by standard methods like MEM
  • ω → 0 extrapolation
  • Non-zero mass
  • Nf = 2
  • β corrections

Marc Klegrewe Temporal Correlators in the Continuous Time Limit of Strong Coupling Lattice QCD 14 / 14