Random Matrix Theory

Transcription

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2 Random Matrix Theory Gernot Akemann Faculty of Physics, Bielefeld University STRONGnet summer school, ZiF Bielefeld, June 2011

3 Content What is RMT about? Nuclear Physics, Number Theory, Quantum Chaos,... quantum counterparts of classically chaotic systems display random matrix statistics [BGS 84] 2/18

4 Content What is RMT about? Nuclear Physics, Number Theory, Quantum Chaos,... quantum counterparts of classically chaotic systems display random matrix statistics [BGS 84] What does RMT have to do with QCD? describes spectrum of QCD Dirac-operator in precise limit (N c = 3): ε-regime of χpt(un-physical finite-v limit) RMT to LO 2/18

5 Content What is RMT about? Nuclear Physics, Number Theory, Quantum Chaos,... quantum counterparts of classically chaotic systems display random matrix statistics [BGS 84] What does RMT have to do with QCD? describes spectrum of QCD Dirac-operator in precise limit (N c = 3): ε-regime of χpt(un-physical finite-v limit) RMT to LO What can we compute in RMT? - analytic form of spectral density, distribution of smallest eigenvalue - as functions of: Σ = qq, F (if we include µ iso ), χsb pattern, gauge field topology = ν zero modes of D (index Theorem) 2/18

6 Content What is RMT about? Nuclear Physics, Number Theory, Quantum Chaos,... quantum counterparts of classically chaotic systems display random matrix statistics [BGS 84] What does RMT have to do with QCD? describes spectrum of QCD Dirac-operator in precise limit (N c = 3): ε-regime of χpt(un-physical finite-v limit) RMT to LO What can we compute in RMT? - analytic form of spectral density, distribution of smallest eigenvalue - as functions of: Σ = qq, F (if we include µ iso ), χsb pattern, gauge field topology = ν zero modes of D (index Theorem) What cannot be computed? - actual value of Σ or F : fit density[σ, F] to Lattice data - χsb or not: β(n f ) 2/18

7 Example: Lattice vs spectrum of Dirac eigenvalue density 0.5 density ρ D (λ) Trδ(D λ) QCD lim 1 V Σ ρ(x) = x 2 [J 0(x 2 ) + J 1 (x 2 )] 1st eigenvalue p 1 (x) = x 2 e x2 /4 at ν = 0, with λv Σ = x spacing 1/V 1/L free case 3/18

8 Example: Lattice vs spectrum of Dirac eigenvalue density 0.5 density ρ D (λ) Trδ(D λ) QCD lim 1 V Σ ρ(x) = x 2 [J 0(x 2 ) + J 1 (x 2 )] 1st eigenvalue p 1 (x) = x 2 e x2 /4 at ν = 0, with λv Σ = x spacing 1/V 1/L free case chue chse choe P min (ζ) 0.2 ρ S (ζ) ζ ζ 3/18

9 Lattice vs. smallest Dirac eigenvalues 1st Dirac-eigenvalue vs. Lattice with chiral fermions [Edwards et at. 98]: 4/18

10 Lattice vs. smallest Dirac eigenvalues 1st Dirac-eigenvalue vs. Lattice with chiral fermions [Edwards et at. 98]: ν = 0 (top): SU(2) SU(3) SU(2) adj. & ν = 1 (bottom): different χsb patters 4/18

11 Lecture 1 the approximation of QCD εχpt equivalence εχpt to RMT and limitations first results from RMT: - flavor-topology duality and - smallest eigenvalue distribution 5/18

12 Setup QCD full theory (Euclidean) +ε µνρσ F µν F ρσ topological term Z QCD [da][dψ] exp[ Tr Ψ(/D + M)Ψ + FF + iθf F] with N f quarks Ψ of masses M, gauge fields A and field strength F 6/18

13 Setup QCD full theory (Euclidean) +ε µνρσ F µν F ρσ topological term Z QCD [da][dψ] exp[ Tr Ψ(/D + M)Ψ + FF + iθf F] with N f quarks Ψ of masses M, gauge fields A and field strength F integrate out quarks Z QCD Nf ν [da]ν f=1 det[d + m f] exp[ Tr F 2 + iθν] 6/18

14 Setup QCD full theory (Euclidean) +ε µνρσ F µν F ρσ topological term Z QCD [da][dψ] exp[ Tr Ψ(/D + M)Ψ + FF + iθf F] with N f quarks Ψ of masses M, gauge fields A and field strength F integrate out quarks Z QCD Nf ν [da]ν f=1 det[d + m f] exp[ Tr F 2 + iθν] properties of the Dirac operator: - diagonalise: Dψ k = iλ k ψ k in finite V, Euclidean D = D - index theorem: ν # zero modes winding # of A - sum over topology Z QCD ν Z ν exp[iθν] - {D, γ 5 } = 0: eigenvalues in pairs ±iλ k 6/18

15 Setup QCD full theory (Euclidean) +ε µνρσ F µν F ρσ topological term Z QCD [da][dψ] exp[ Tr Ψ(/D + M)Ψ + FF + iθf F] with N f quarks Ψ of masses M, gauge fields A and field strength F integrate out quarks Z QCD Nf ν [da]ν f=1 det[d + m f] exp[ Tr F 2 + iθν] properties of the Dirac operator: - diagonalise: Dψ k = iλ k ψ k in finite V, Euclidean D = D - index theorem: ν # zero modes winding # of A - sum over topology Z QCD ν Z ν exp[iθν] - {D, γ 5 } = 0: eigenvalues in pairs ±iλ k formally Z QCD = ν 0 [dλ] N f f=1 mν f eiνθ/n f k (λ2 k + m2 f ) exp[ Tr F 2 [λ]] 6/18

16 Start: chiral perturbation theory χpt for QCD Integration of all non-goldstone modes: in a box V = L 4 : valid for momenta 1/L Λ non-goldstone scale Z χpt SU(N f ) [du(x)] exp[ dxtr L(U, U)] expansion in higher orders L = L 2 + L 4 + L ) L 2 = 1 4 F 2 α U(x) α U (x) (e ΣM i Θ N f U(x) + e i Θ N f U (x) LECs: Pion decay constant F & chiral condensate Σ M = diag(m u, m d,...) quark masses 7/18

17 Start: chiral perturbation theory χpt for QCD Integration of all non-goldstone modes: in a box V = L 4 : valid for momenta 1/L Λ non-goldstone scale Z χpt SU(N f ) [du(x)] exp[ dxtr L(U, U)] expansion in higher orders L = L 2 + L 4 + L ) L 2 = 1 4 F 2 α U(x) α U (x) (e ΣM i Θ N f U(x) + e i Θ N f U (x) LECs: Pion decay constant F & chiral condensate Σ M = diag(m u, m d,...) quark masses standard expansion U(x) = exp[iπ k (x)σ k /F], Pion mass: m 2 πf 2 = Σ N f f=1 m f GOR relation 7/18

18 Start: chiral perturbation theory χpt for QCD Integration of all non-goldstone modes: in a box V = L 4 : valid for momenta 1/L Λ non-goldstone scale Z χpt SU(N f ) [du(x)] exp[ dxtr L(U, U)] expansion in higher orders L = L 2 + L 4 + L ) L 2 = 1 4 F 2 α U(x) α U (x) (e ΣM i Θ N f U(x) + e i Θ N f U (x) LECs: Pion decay constant F & chiral condensate Σ M = diag(m u, m d,...) quark masses standard expansion U(x) = exp[iπ k (x)σ k /F], Pion mass: m 2 πf 2 = Σ N f f=1 m f GOR relation Fourier trafo: fix topology dθe iνθ SU(N f ) du = U(N f ) du det[u]ν 7/18

19 The ε-regime ε counting [Gasser, Leutwyler 87] m π 1 L 2 1 L = ε = V 4 (m 2 πv ) 1 = O(1) unphysical regime!! fig: [Bietenholz et al. physics/ ] 8/18

20 The ε-regime ε counting [Gasser, Leutwyler 87] m π 1 L 2 1 L = ε = V 4 (m 2 πv ) 1 = O(1) unphysical regime!! fig: [Bietenholz et al. physics/ ] BUT: we can do analytic, non-perturbative calculations 8/18

21 The ε-regime ε counting [Gasser, Leutwyler 87] m π 1 L 2 1 L = ε = V 4 (m 2 πv ) 1 = O(1) unphysical regime!! fig: [Bietenholz et al. physics/ ] BUT: we can do analytic, non-perturbative calculations standard p regime counting: m π 1/L 8/18

22 Partition function of εχpt- zero mode dominance fluctuations from integrating quadratic term π k (p 2 + m 2 π)π k : ( ) O small for p 0 NOT for p = 0 1 V (p 2 +m 2 π) 9/18

23 Partition function of εχpt- zero mode dominance fluctuations from integrating quadratic term π k (p 2 + m 2 π)π k : ( ) O small for p 0 NOT for p = 0 1 V (p 2 +m 2 π) zero-momentum mode dominate: non-perturbative treatment parametrise U(x) = U 0 e iπ(x)/f ε-regime: L 2 exact & factorisation: Z ν,εχpt = U(N f ) du 0 det[u 0 ] ν e 1 2 ΣV Tr( M(U 0 +U 0 ) ) [dπ]e dx 1 2 ( π)2 9/18

24 Partition function of εχpt- zero mode dominance fluctuations from integrating quadratic term π k (p 2 + m 2 π)π k : ( ) O small for p 0 NOT for p = 0 1 V (p 2 +m 2 π) zero-momentum mode dominate: non-perturbative treatment parametrise U(x) = U 0 e iπ(x)/f ε-regime: L 2 exact & factorisation: Z ν,εχpt = U(N f ) du 0 det[u 0 ] ν e 1 2 ΣV Tr( M(U 0 +U 0 ) ) [dπ]e dx 1 2 ( π)2 Z ν,εχpt det 1,...,N f [ ] ˆm j 1 k I ν+j 1 ( ˆm k ) / Nf ( ˆm 2 f ) [Brower,Tan Rossi 81, etc], with ˆm = mv Σ, = Vandermonde 9/18

25 Partition function of εχpt- zero mode dominance fluctuations from integrating quadratic term π k (p 2 + m 2 π)π k : ( ) O small for p 0 NOT for p = 0 1 V (p 2 +m 2 π) zero-momentum mode dominate: non-perturbative treatment parametrise U(x) = U 0 e iπ(x)/f ε-regime: L 2 exact & factorisation: Z ν,εχpt = U(N f ) du 0 det[u 0 ] ν e 1 2 ΣV Tr( M(U 0 +U 0 ) ) [dπ]e dx 1 2 ( π)2 Z ν,εχpt det 1,...,N f [ ] ˆm j 1 k I ν+j 1 ( ˆm k ) / Nf ( ˆm 2 f ) [Brower,Tan Rossi 81, etc], with ˆm = mv Σ, = Vandermonde group integral part equivalent to RMT to LO: O(ε 0 ) 9/18

26 Limitations of RMT and corrections from εχpt ρ(λ) 3 ~V λ λ min m c Λ QCD chrmt - log λ fig: [Verbaarschot] breakdown at Thouless Energy λ c F 2 / V [Osborn, Verbaarschot 01] 10/18

27 Limitations of RMT and corrections from εχpt ρ(λ) 3 ~V λ λ min m c Λ QCD chrmt - log λ fig: [Verbaarschot] breakdown at Thouless Energy λ c F 2 / V [Osborn, Verbaarschot 01] corrections to RMT regime can be computed: 1-loop εχpt ( ) in Z corrections Σ eff = Σ 1 N2 f 1 (0) N f F, F eff =... in 1 V [Gasser, Leutwyler 87; G.A., Basile, Lellouch 08] spectral D-correlations from RMT with eff. LEC 10/18

28 Limitations of RMT and corrections from εχpt ρ(λ) 3 ~V λ λ min m c Λ QCD chrmt - log λ fig: [Verbaarschot] breakdown at Thouless Energy λ c F 2 / V [Osborn, Verbaarschot 01] corrections to RMT regime can be computed: 1-loop εχpt ( ) in Z corrections Σ eff = Σ 1 N2 f 1 (0) N f F, F eff =... in 1 V [Gasser, Leutwyler 87; G.A., Basile, Lellouch 08] spectral D-correlations from RMT with eff. LEC for Green s function π(x)π(0) : need εχpt[hansen 90, Damgaard et al. 01] 10/18

29 What can we learn from εχpt? LO: non-zero modes = Gaussian free fields overall constant 11/18

30 What can we learn from εχpt? LO: non-zero modes = Gaussian free fields overall constant study quark mass dependence of partition function: - Taylor expand in m q formal expression for Z QCD in terms of λ k Z ν ({m q }) = 0 [dλ] N f f=1 mν f k (λ2 k + m2 f ) exp[ Tr F 2 [λ]] - compare to group integral [ ] Z ν,εχpt det 1,...,N f ˆm j 1 k I ν+j 1 ( ˆm k ) / Nf ( ˆm 2 f ) 11/18

31 What can we learn from εχpt? LO: non-zero modes = Gaussian free fields overall constant study quark mass dependence of partition function: - Taylor expand in m q formal expression for Z QCD in terms of λ k Z ν ({m q }) = 0 [dλ] N f f=1 mν f k (λ2 k + m2 f ) exp[ Tr F 2 [λ]] - compare to group integral [ ] Z ν,εχpt det 1,...,N f ˆm j 1 k I ν+j 1 ( ˆm k ) / Nf ( ˆm 2 f ) Leutwyler-Smilga sum rules: ) m (1 ν + m 2 k 1 λ k QCD 1 k V Σλ = 1 2 k QCD 4(ν+1) = I ν (m) 11/18

32 What can we learn from εχpt? LO: non-zero modes = Gaussian free fields overall constant study quark mass dependence of partition function: - Taylor expand in m q formal expression for Z QCD in terms of λ k Z ν ({m q }) = 0 [dλ] N f f=1 mν f k (λ2 k + m2 f ) exp[ Tr F 2 [λ]] - compare to group integral [ ] Z ν,εχpt det 1,...,N f ˆm j 1 k I ν+j 1 ( ˆm k ) / Nf ( ˆm 2 f ) Leutwyler-Smilga sum rules: ) m (1 ν + m 2 k 1 λ k QCD 1 k V Σλ = 1 2 k QCD 4(ν+1) = I ν (m) computation of spectral 1-point and 2-point density possible by adding extra source quarks (fermionic and bosonic) next lecture 11/18

33 The Matrix Model of QCD in RMT computation of ALL spectral densities and individual eigenvalue distributions possible and much simpler 12/18

34 The Matrix Model of QCD in RMT computation of ALL spectral densities and individual eigenvalue distributions possible and much simpler Z ν, RMT m f iw dw Π N f f=1 det e NΣ2 TrWW iw [Shuryak, Verbaarschot 93] m f block matrix has same global symmetry as QCD-Dirac operator D: W ij C: N (N + ν) Gaussian random variables (chgue) 2N + ν = V eigenvalues: N ±λ k, ν zero eigenvalues 12/18

35 The Matrix Model of QCD in RMT computation of ALL spectral densities and individual eigenvalue distributions possible and much simpler Z ν, RMT m f iw dw Π N f f=1 det e NΣ2 TrWW iw [Shuryak, Verbaarschot 93] m f block matrix has same global symmetry as QCD-Dirac operator D: W ij C: N (N + ν) Gaussian random variables (chgue) 2N + ν = V eigenvalues: N ±λ k, ν zero eigenvalues universal large N V (volume) limit: lim N Z ν,rmt = Z ν,εχpt 12/18

36 The Matrix Model of QCD in RMT computation of ALL spectral densities and individual eigenvalue distributions possible and much simpler Z ν, RMT m f iw dw Π N f f=1 det e NΣ2 TrWW iw [Shuryak, Verbaarschot 93] m f block matrix has same global symmetry as QCD-Dirac operator D: W ij C: N (N + ν) Gaussian random variables (chgue) 2N + ν = V eigenvalues: N ±λ k, ν zero eigenvalues universal large N V (volume) limit: lim N Z ν,rmt = Z ν,εχpt equivalence of other objects than Z? - all Dirac eigenvalue correlations [Damgaard et al 01, Basile, G.A. 07] 12/18

37 The Matrix Model of QCD in RMT computation of ALL spectral densities and individual eigenvalue distributions possible and much simpler Z ν, RMT m f iw dw Π N f f=1 det e NΣ2 TrWW iw [Shuryak, Verbaarschot 93] m f block matrix has same global symmetry as QCD-Dirac operator D: W ij C: N (N + ν) Gaussian random variables (chgue) 2N + ν = V eigenvalues: N ±λ k, ν zero eigenvalues universal large N V (volume) limit: lim N Z ν,rmt = Z ν,εχpt equivalence of other objects than Z? - all Dirac eigenvalue correlations [Damgaard et al 01, Basile, G.A. 07] different εχpt-χsb, classes: W ij R, H: RMT still solvable 12/18

38 Large-N limit and equivalence to εχpt Σ = 1 for simplicity: Z ν,rmt = dw N f f det[ww + m 2 WW f ]e NTr 13/18

39 Large-N limit and equivalence to εχpt Σ = 1 for simplicity: Z ν,rmt = dw N f f det[ww + m 2 WW f ]e NTr step 1. Grassmann det[...] = dψ exp[ψ... ψ] do Gauß dw 13/18

40 Large-N limit and equivalence to εχpt Σ = 1 for simplicity: Z ν,rmt = dw N f f det[ww + m 2 WW f ]e NTr step 1. Grassmann det[...] = dψ exp[ψ... ψ] do Gauß dw step 2. ψ 4 Hubbard-Stratonovich: extra dq Nf N f do dψ: Z ν,rmt dqdet[q ] ν det [ Q Q ] N e NTr(Q Q M(Q+Q )) 13/18

41 Large-N limit and equivalence to εχpt Σ = 1 for simplicity: Z ν,rmt = dw N f f det[ww + m 2 WW f ]e NTr step 1. Grassmann det[...] = dψ exp[ψ... ψ] do Gauß dw step 2. ψ 4 Hubbard-Stratonovich: extra dq Nf N f do dψ: Z ν,rmt dqdet[q ] ν det [ Q Q ] N e NTr(Q Q M(Q+Q )) step 3. Saddle Point N : parametrise Q = UR, at SP: R 1, + scale masses Nm f Z ν,rmt du U(Nf ) det[u ] ν e NTrM(U+U ) 0-dimensional σ-model 13/18

42 RMT eigenvalue representation Z ν,rmt dw Π N f f=1 det m f iw iw m f e NTrW jw j diagonalise WW positive definite matrix : eigenvalues λ k 0 (or singular values of W : y k with y 2 k = λ k Dirac eigenvalues Z ν,rmt 0 N k dλ kλ ν k e NΣ2 λ k Nf f (λ k + m 2 f ) N(λ) 2 integrand P jpdf joint probability distribution 14/18

43 RMT eigenvalue representation Z ν,rmt dw Π N f f=1 det m f iw iw m f e NTrW jw j diagonalise WW positive definite matrix : eigenvalues λ k 0 (or singular values of W : y k with y 2 k = λ k Dirac eigenvalues Z ν,rmt 0 N k dλ kλ ν k e NΣ2 λ k Nf f (λ k + m 2 f ) N(λ) 2 integrand P jpdf joint probability distribution Vandermonde determinant N (λ) = k>l (λ k λ l ) = exp[2 k>l log[λ k λ l ]] eigenvalues strongly coupled: log-gas, repulsion of eigenvalues 14/18

44 Results from RMT exhibits flavour-topology duality [Verbarschot]: lim m 0 Z (N f) ν /m ν Z (N f 1) ν+1 - this can also be shown on the level of the εχptpartition function [ADDV 01] 15/18

45 Results from RMT exhibits flavour-topology duality [Verbarschot]: lim m 0 Z (N f) ν /m ν Z (N f 1) ν+1 - this can also be shown on the level of the εχptpartition function [ADDV 01] e.g. two massless flavours at ν = 0 = quenched n f = 2 at ν = 2 valid only to LO in the epsilon regime!! note that in general we take the chiral limit as lim m 0, V mv Σ finite 15/18

46 Results from RMT exhibits flavour-topology duality [Verbarschot]: lim m 0 Z (N f) ν /m ν Z (N f 1) ν+1 - this can also be shown on the level of the εχptpartition function [ADDV 01] e.g. two massless flavours at ν = 0 = quenched n f = 2 at ν = 2 valid only to LO in the epsilon regime!! note that in general we take the chiral limit as lim m 0, V mv Σ finite define spectral density of Dirac eigenvalues y = λ: insert δ ρ(y) = Trδ(D y) = N δ(y 1 y) dy 2... dy N P jpdf ({y}) - can be computed exactly for finite-n 15/18

47 Exercise 1st eigenvalue distribution of the smallest eigenvalue p 1 (y): from gap probability = probability that all eigenvalues are > λ: E(λ) = 1 Z λ N N f dλ k λ ν λ k k e NΣ2 (λ k + m 2 f ) N(λ) 2 k=1 f λ E(λ) is the probability that 1 eigenvalue is at λ and all others are λ 16/18

48 Exercise 1st eigenvalue distribution of the smallest eigenvalue p 1 (y): from gap probability = probability that all eigenvalues are > λ: E(λ) = 1 Z λ N N f dλ k λ ν λ k k e NΣ2 (λ k + m 2 f ) N(λ) 2 k=1 f λ E(λ) is the probability that 1 eigenvalue is at λ and all others are λ Exercise: check for ν = 0 that the following holds for Dirac eigenvalues 1) for N f = 0 derive p 1 (x) = x/2e x2 /4 with x = 2NΣy 2) derive p 1 (x) for arbitrary N f > 0 16/18

49 Some literature H. Leutwyler, A. Smilga Phys Rev D46 (1992) E. Shuryak, J. Verbaarschot, Random matrix theory and spectral sum rules for the Dirac operator in QCD hep-th/ = Nucl.Phys.A560: ,1993 Jacobus Verbaarschot The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way, hep-th/ = Phys.Rev.Lett. 72 (1994) G.A., P. H. Damgaard, U. Magnea, S. Nishigaki Universality of random matrices in the microscopic limit and the Dirac operator spectrum hep-th/ = Nucl.Phys.B487: , /18

50 Lecture 2 spectral statistics from εχptand from RMT inclusion of chemical potential - why χptis sensitive to baryonic µ further extensions: Wilson χpt 18/18