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

Taylor expansion and the Cauchy Residue Theorem for finite-density QCD Benjamin Jäger In collaboration with Philippe de Forcrand (ETH Zürich)

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Phase diagram for QCD Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Taylor Expansion • Expand around small chemical potentials µ � µ � k P ( µ , t ) � = c k ( T ) , k = 0, 2, ... T 4 T k • The Taylor coefficients can computed at µ = 0 ∂ k log Z � 1 � c k = � n ! VT 3 ∂ ( µ/ T ) k � � µ =0 • Typical building blocks � M − 1 ∂ 2 M ∂µ 2 M − 5 ∂ 4 M � M − 1 ∂ M � � Tr , ... , Tr ∂µ 4 ∂µ • Use linear chemical potential to reduce to single form [Gavai & Sharma, 2014] �� � k � M − 1 ∂ M Tr ∂µ • Estimate traces using many noise vectors Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Spectrum Staggered quarks: 4 · 4 3 , N f = 4, β = 5.05, m = 0.07 Hermitian matrix Non-Hermitian matrix D − 1 ∂ / D i / / D ∂µ • Real Eigenvalues • Complex Eigenvalues 0.2 6 4 0 2 Im λ M Im λ M − 0.2 0.2 0 − 2 − 0.4 0 − 4 − 0.2 − 0.6 − 6 0 0.1 0.2 0.3 0.4 0.5 − 4 − 2 0 2 4 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Chebyshev Polynomials 1.2 Chebyshev Polynomials p = 50 1 p = 200 T 0 ( x ) = 1, T 1 ( x ) = x 0.8 T n +1 ( x ) = 2 xT n ( x ) − T n − 1 ( x ) 0.6 • Construct arbitrary function 0.4 n � f ( x ) = γ p T p ( x ) 0.2 p =0 0 • Can be extended to matrix func. − 0.2 0 0.2 0.4 0.6 0.8 1 • x ∈ [ − 1, 1] x Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Chebyshev Polynomials 800 424 700 Eigenvalues count n [ − 1, x ] 422 600 420 500 418 400 0.19 0.2 0.21 300 200 Exact 100 Chebyshev 0 − 1 − 0.5 0 0.5 1 x • Number of eigenvalues in the interval n [ − 1, x ] p max n V Giusti & Lüscher 2008, n [ a , b ] = 1 � � g p ( a , b ) � η † Fodor et. al. 2016, i T p ( A ) η i � Cossu et. al. 2016 n i =1 p =0 Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Chebyshev Polynomials Tr( D − k ) 10 3 n V = 1, k = 2 n V = 1, k = 8 10 2 10 1 |relative error| 10 0 10 − 1 10 − 2 10 − 3 10 0 10 1 10 2 10 3 10 4 10 5 p - order of polynomial � � � exact − approx. • Absolute relative error, i.e. � � exact � Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Chebyshev Polynomials Tr( D − k ) 10 1 n V = 2, p = 4096 n V = 2, p = 32768 10 0 |relative error| 10 − 1 10 − 2 10 − 3 0 5 10 15 20 k - moment � D − k � • Accuracy of Tr / for larger moments (or cumulants) Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Chebyshev Polynomials 1.2 p = 50 1 p = 200 Chebyshev Polynomials 0.8 • Advantages • No inversion necessary 0.6 • Good accuracy on eigenvalues 0.4 • Disadvantages 0.2 • Only works for Hermitian matrix • Limited applicability 0 − 0.2 0 0.2 0.4 0.6 0.8 1 x Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Chebyshev Polynomials 1.2 p = 50 1 Chebyshev Polynomials p = 200 • Advantages 0.8 • No inversion necessary 0.6 • Good accuracy on eigenvalues • Disadvantages 0.4 • Only works for Hermitian matrix 0.2 • Limited applicability 0 Can we do better? − 0.2 0 0.2 0.4 0.6 0.8 1 x Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Cauchy Residue Theorem 1 � f ( z ) d z = Res [ f ( z )] 2 π i Γ • Number of eigenvalues in the contour µ ( Γ ) 1 � � ( A − z 1 ) − 1 � µ ( Γ ) = Tr d z 2 π i Γ • Use inverse matrix (spacewise sparse) and shifted solver � − 1 � D − 1 ∂ / ∂ / D D A = M − 1 = / M = / D ∂µ ∂µ • Start with larger box: Compute contour and refine (# λ > 1) Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Cauchy Residue Theorem 1 � f ( z ) d z = Res [ f ( z )] 2 π i Γ • Number of eigenvalues in discrete contour µ ( Γ ) n Q µ ( Γ ) ∼ 1 � ( A − z k 1 ) − 1 � � z k Tr n Q k =0 • Use inverse matrix (spacewise sparse) and shifted solver � − 1 � D − 1 ∂ / ∂ / D D A = M − 1 = / M = / D ∂µ ∂µ • Start with larger box: Discretize contour and refine (# λ > 1) Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Eigenvalues & Refinement 6 4 2 Im λ M 0 − 2 − 4 − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Eigenvalues & Refinement 6 4 2 Im λ M 0 − 2 − 4 − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Eigenvalues & Refinement 6 4 2 Im λ M 0 − 2 − 4 − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Eigenvalues & Refinement 6 4 2 Im λ M 0 − 2 − 4 − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Eigenvalues & Refinement 6 4 2 Im λ M 0 − 2 − 4 − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Eigenvalues & Refinement 6 4 2 Im λ M 0 − 2 − 4 − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Eigenvalues & Refinement 6 4 2 Im λ M 0 − 2 − 4 − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Eigenvalues & Refinement 6 4 2 Im λ M 0 − 2 − 4 − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Accuracy Tr( M − k ) 10 3 k = 4 10 2 k = 8 10 1 |relative error| 10 0 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 10 1 10 2 l - size of box � � � exact − approx. • Absolute relative error, i.e. � � exact � Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Accuracy Tr( M − k ) 10 2 box size 0.0037 10 1 box size 0.0009 10 0 |relative error| 10 − 1 10 − 2 10 − 3 10 − 4 10 − 5 0 5 10 15 20 k - moment Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Cauchy Residue Theorem I 6 4 Refinement procedure 2 • Advantages Im λ M 0 • Very good accuracy • Even for larger moments − 2 • Disadvantages − 4 • A lot of shifted inversions necessary − 6 − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Cauchy Residue Theorem I 6 4 Refinement procedure • Advantages 2 Im λ M • Very good accuracy 0 • Even for larger moments • Disadvantages − 2 • A lot of shifted inversions necessary − 4 − 6 Can we do better? − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M Benjamin Jäger Lattice 2018 27.07.2017

Introduction Chebyshev Polynomials Cauchy Residue Theorem I Cauchy Residue Theorem II Cauchy Residue Theorem II Use a single discrete circular contour • Number of eigenvalues 6 n Q µ ( Γ ) ∼ 1 � ( M − z k 1 ) − 1 � � z k Tr 4 n Q k =0 2 • Γ : Circle containing no eigenvalues Im λ M 0 2 π i N k z k = r e − 2 • Use inverse moments − 4 �� / �� / D ′ / − 6 D − 1 � k � D ′− 1 � − k � D / Tr = Tr − 4 − 3 − 2 − 1 0 1 2 3 4 Re λ M ✓ ✏ N �� / ∼ 1 D ′− 1 � − k � D ′− 1 � − 1 � �� z i 1 − / � z − k D / D / Tr Tr i N ✒ i =1 ✑ Benjamin Jäger Lattice 2018 27.07.2017