Taylor expansion and the Cauchy Residue Theorem for finite-density - - PowerPoint PPT Presentation

▶
taylor expansion and the cauchy residue theorem for
SMART_READER_LITE
LIVE PREVIEW

Taylor expansion and the Cauchy Residue Theorem for finite-density - - PowerPoint PPT Presentation

Dec 19, 2022 235 likes •583 views

Taylor expansion and the Cauchy Residue Theorem for finite-density QCD Benjamin Jger In collaboration with Philippe de Forcrand (ETH Zrich) Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Phase

slide-1
SLIDE 1

Taylor expansion and the Cauchy Residue Theorem for finite-density QCD Benjamin Jäger

In collaboration with Philippe de Forcrand (ETH Zürich)

slide-2
SLIDE 2

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Phase diagram for QCD

Benjamin Jäger Lattice 2018 27.07.2017

slide-3
SLIDE 3

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Taylor Expansion

  • Expand around small chemical potentials µ

P(µ, t) T 4 =

  • k

ck(T)

µ

T

k

, k = 0, 2, ...

  • The Taylor coefficients can computed at µ = 0

ck = 1 n! VT 3 ∂k log Z ∂ (µ/T)k

  • µ=0
  • Typical building blocks

Tr

  • M−1 ∂M

∂µ

  • , ... , Tr
  • M−1 ∂2M

∂µ2 M−5 ∂4M ∂µ4

  • Use linear chemical potential to reduce to single form

Tr

  • M−1 ∂M

∂µ

k

  • Estimate traces using many noise vectors

Benjamin Jäger Lattice 2018 27.07.2017 [Gavai & Sharma, 2014]

slide-4
SLIDE 4

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Spectrum

Hermitian matrix i / D

  • Real Eigenvalues

Non-Hermitian matrix / D−1 ∂ / D ∂µ

  • Complex Eigenvalues

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −0.6 −0.4 −0.2 0.2 −4 −2 2 4 −0.2 0.2 0.1 0.2 0.3 0.4 0.5 Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 0 1 2 3 4 Staggered quarks: 4 · 43, Nf = 4, β = 5.05, m = 0.07

slide-5
SLIDE 5

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Chebyshev Polynomials

Chebyshev Polynomials T0(x) = 1, T1(x) = x Tn+1(x) = 2 xTn(x) − Tn−1(x)

  • Construct arbitrary function

f (x) =

n

  • p=0

γp Tp(x)

  • Can be extended to matrix func.
  • x ∈ [−1, 1]

Benjamin Jäger Lattice 2018 27.07.2017

x p = 50 p = 200 −0.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1

slide-6
SLIDE 6

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Chebyshev Polynomials

  • Number of eigenvalues in the interval n[−1, x]

n[a, b] = 1 n

nV

  • i=1

pmax

  • p=0

gp(a, b) η†

i Tp(A) ηi

Benjamin Jäger Lattice 2018 27.07.2017

Eigenvalues count n[−1, x] x Exact Chebyshev 100 200 300 400 500 600 700 800 −1 −0.5 0.5 1 418 420 422 424 0.19 0.2 0.21

Giusti & Lüscher 2008, Fodor et. al. 2016, Cossu et. al. 2016

slide-7
SLIDE 7

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Chebyshev Polynomials

  • Absolute relative error, i.e.
  • exact−approx.

exact

  • Benjamin Jäger

Lattice 2018 27.07.2017

Tr(D−k) |relative error| p - order of polynomial nV = 1, k = 2 nV = 1, k = 8 10−3 10−2 10−1 100 101 102 103 100 101 102 103 104 105

slide-8
SLIDE 8

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Chebyshev Polynomials

  • Accuracy of Tr
  • /

D−k for larger moments (or cumulants)

Benjamin Jäger Lattice 2018 27.07.2017

Tr(D−k) |relative error| k - moment nV = 2, p = 4096 nV = 2, p = 32768 10−3 10−2 10−1 100 101 5 10 15 20

slide-9
SLIDE 9

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Chebyshev Polynomials

Chebyshev Polynomials

  • Advantages
  • No inversion necessary
  • Good accuracy on eigenvalues
  • Disadvantages
  • Only works for Hermitian matrix
  • Limited applicability

Benjamin Jäger Lattice 2018 27.07.2017

x p = 50 p = 200 −0.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1

slide-10
SLIDE 10

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Chebyshev Polynomials

Chebyshev Polynomials

  • Advantages
  • No inversion necessary
  • Good accuracy on eigenvalues
  • Disadvantages
  • Only works for Hermitian matrix
  • Limited applicability

Can we do better?

Benjamin Jäger Lattice 2018 27.07.2017

x p = 50 p = 200 −0.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1

slide-11
SLIDE 11

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Cauchy Residue Theorem

  • Number of eigenvalues in the contour µ(Γ)

µ(Γ) = 1 2πi

  • Γ

Tr

  • (A − z1)−1

dz

  • Use inverse matrix (spacewise sparse) and shifted solver

M = / D−1 ∂ / D ∂µ A = M−1 = / D

  • ∂ /

D ∂µ

−1

  • Start with larger box: Compute contour and refine (#λ > 1)

Benjamin Jäger Lattice 2018 27.07.2017

1 2πi

  • Γ

f (z) dz = Res [f (z)]

slide-12
SLIDE 12

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Cauchy Residue Theorem

  • Number of eigenvalues in discrete contour µ(Γ)

µ(Γ) ∼ 1 nQ

nQ

  • k=0

zkTr

  • (A − zk1)−1
  • Use inverse matrix (spacewise sparse) and shifted solver

M = / D−1 ∂ / D ∂µ A = M−1 = / D

  • ∂ /

D ∂µ

−1

  • Start with larger box: Discretize contour and refine (#λ > 1)

Benjamin Jäger Lattice 2018 27.07.2017

1 2πi

  • Γ

f (z) dz = Res [f (z)]

slide-13
SLIDE 13

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Eigenvalues & Refinement

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 1 2 3 4

slide-14
SLIDE 14

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Eigenvalues & Refinement

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 1 2 3 4

slide-15
SLIDE 15

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Eigenvalues & Refinement

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 1 2 3 4

slide-16
SLIDE 16

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Eigenvalues & Refinement

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 1 2 3 4

slide-17
SLIDE 17

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Eigenvalues & Refinement

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 1 2 3 4

slide-18
SLIDE 18

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Eigenvalues & Refinement

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 1 2 3 4

slide-19
SLIDE 19

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Eigenvalues & Refinement

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 1 2 3 4

slide-20
SLIDE 20

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Eigenvalues & Refinement

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 1 2 3 4

slide-21
SLIDE 21

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Accuracy

  • Absolute relative error, i.e.
  • exact−approx.

exact

  • Benjamin Jäger

Lattice 2018 27.07.2017

Tr(M−k) |relative error| l - size of box k = 4 k = 8 10−4 10−3 10−2 10−1 100 101 102 103 10−5 10−4 10−3 10−2 10−1 100 101 102

slide-22
SLIDE 22

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Accuracy

Benjamin Jäger Lattice 2018 27.07.2017

Tr(M−k) |relative error| k - moment box size 0.0037 box size 0.0009 10−5 10−4 10−3 10−2 10−1 100 101 102 5 10 15 20

slide-23
SLIDE 23

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Cauchy Residue Theorem I

Refinement procedure

  • Advantages
  • Very good accuracy
  • Even for larger moments
  • Disadvantages
  • A lot of shifted inversions necessary

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 0 1 2 3 4

slide-24
SLIDE 24

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Cauchy Residue Theorem I

Refinement procedure

  • Advantages
  • Very good accuracy
  • Even for larger moments
  • Disadvantages
  • A lot of shifted inversions necessary

Can we do better?

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 0 1 2 3 4

slide-25
SLIDE 25

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Cauchy Residue Theorem II

Use a single discrete circular contour

  • Number of eigenvalues

µ(Γ) ∼ 1 nQ

nQ

  • k=0

zkTr

  • (M − zk1)−1
  • Γ: Circle containing no eigenvalues

zk = r e

2πi N k

  • Use inverse moments

Tr

/

D′ / D−1k = Tr

/

D / D′−1−k

✓ ✒ ✏ ✑

Tr

/

D / D′−1−k ∼ 1 N

N

  • i=1

z−k

i

Tr

zi1 − /

D / D′−1−1

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 0 1 2 3 4

slide-26
SLIDE 26

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Accuracy

  • Absolute relative error, i.e.
  • exact−approx.

exact

  • Benjamin Jäger

Lattice 2018 27.07.2017

Tr(M−k) |relative error| nQ - number of quadrature points nV = 8, k = 2 nV = 8, k = 8 10−4 10−3 10−2 10−1 100 101 102 100 101 102 103 104

slide-27
SLIDE 27

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Accuracy

Benjamin Jäger Lattice 2018 27.07.2017

Tr(M−k) |relative error| nV - number of noise vectors nQ = 4096, k = 2 nQ = 4096, k = 8 10−4 10−3 10−2 10−1 100 101 102 20 40 60 80 100 120 140

slide-28
SLIDE 28

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Accuracy

Benjamin Jäger Lattice 2018 27.07.2017

λmin Tr(M−k) |relative error| r - radius k = 2, nV = 8, nQ = 4096 k = 8, nV = 8, nQ = 4096 10−4 10−3 10−2 10−1 100 101 102 103 0.01 0.1 1 10

slide-29
SLIDE 29

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Accuracy

Benjamin Jäger Lattice 2018 27.07.2017

Tr(M−k) |relative error| k - moment nV = 32, nQ = 2048 nV = 32, nQ = 8192 10−4 10−3 10−2 10−1 100 101 102 103 104 5 10 15 20

slide-30
SLIDE 30

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Accuracy

  • Tr

/

D′ / D−1k scaled by k-th root - relative size

Benjamin Jäger Lattice 2018 27.07.2017

Tr(M−k)(1/k) Tr(M−k)(1/k) k - moment nV = 4, nQ = 256 nV = 4, nQ = 2048 exact 2 4 6 8 10 12 14 5 10 15 20

slide-31
SLIDE 31

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Accuracy

  • Volumes:

4 · 43 4 · 63 4 · 83

Benjamin Jäger Lattice 2018 27.07.2017

Tr(M−k) |relative error| V - volume k = 2, nV = 8, nQ = 1024 k = 8, nV = 8, nQ = 1024 10−4 10−3 10−2 10−1 100 101 1000 10000

slide-32
SLIDE 32

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Cauchy Residue Theorem II

Single contour approach

  • Advantages
  • Good accuracy
  • Even for larger moments
  • Moderate effort

Benjamin Jäger Lattice 2018 27.07.2017

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 0 1 2 3 4

slide-33
SLIDE 33

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Conclusion

Conclusion

  • Based on Cauchy Residue Theorem
  • Good accuracy for large moments
  • Moderate effort

Future Work

  • Truncated solvers or all-mode averaging
  • Block solvers
  • Multishift block solvers [de Forcrand & Keegan]

Benjamin Jäger Lattice 2018 27.07.2017 x p = 50 p = 200 −0.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1

Im λM Re λM −6 −4 −2 2 4 6 −4 −3 −2 −1 0 1 2 3 4

slide-34
SLIDE 34

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II

Questions?

Benjamin Jäger Lattice 2018 27.07.2017