SUPERSYMMETRY WITHOUT SUPERSYMMETRY : DETERMINANTS AND PFAFFIANS IN RMT

Transcription

1 SUPERSYMMETRY WITHOUT SUPERSYMMETRY : DETERMINANTS AND PFAFFIANS IN RMT Mario Kieburg Universität Duisburg-Essen SFB/TR-12 Workshop on Langeoog (Germany), February 22 th 2010 supported by

2 OUTLINE (1) Characteristic polynomials (2) First example: Hermitian matrices (3) Determinantal structures and the connection to supersymmetry (4) Second example: Special orthogonal group (5) Aim of our method and the general theorems (6) General remarks and conclusions

3 Characteristic polynomials

4 CHARACTERISTIC POLYNOMIALS Characteristic polynomial of a N N matrix H is q(x, H) = det(h x), x C PROPERTIES q is a polynomial of order N in x q is invariant under similarity transformations H THT 1 with T invertible roots of q with respect to x are the algebraic eigenvalues {E 1,...,E N } of H q only depends on x and {E 1,...,E N }

5 CHARACTERISTIC POLYNOMIALS Average over rotation invariant matrix ensembles are interesting for: disordered systems quantum chaos matrix models in high energy physics quantum chromodynamics quantum gravity econo physics number theory Weyl s character formula theory of orthogonal polynomials

6 CHARACTERISTIC POLYNOMIALS SETTING Z(κ,λ) k 2 det(h κ n2 ) n=1 P(H) k 1 det(h κ n1 ) n=1 l 2 m=1 l 1 m=1 det(h λ m2 ) d[h] det(h λ m1 ) + factorizable probability density P N P(E) = P(E j ), with E = diag (E 1,...,E N ) j=1 H does not have to be symmetric!

7 First example: Hermitian matrices Herm(N)

8 FIRST EXAMPLE: HERMITIAN MATRICES Herm (N) STARTING POINT Z N (κ) = Herm (N) P(H) k j=1 det(h κ j2 ) det(h κ j1 ) d[h] It is well known that this generating function exhibits a determinantal structure! Baik, Deift, Strahov (2003); Grönqvist, Guhr, Kohler (2004); Borodin, Strahov (2005); Guhr (2006) Similar determinantal structures for the k-point correlation function. Mehta,Gaudin (1960)

9 FIRST EXAMPLE: HERMITIAN MATRICES Herm (N) DEFINITION Ber p/q (s) = = (s a1 s b1 ) (s a2 s b2 ) 1 a<b p 1 a<b q p q (s a1 s b2 ) a=1 b=1 p (s 1 ) q (s 2 ) p q (s a1 s b2 ) a=1 b=1 s = diag (s 1, s 2 ) = diag (s 11,..., s p1, s 12,..., s q2 )

10 FIRST EXAMPLE: HERMITIAN MATRICES Herm (N) RESULT Z N (κ) [ ] 1 ZN (κ a1,κ b2 ) det Ber k/k (κ) κ a1 κ b2 We have not used the explicit form of P(E)! These structures have to be true for other matrix ensembles!

11 Determinantal structures and the connection to supersymmetry

12 DETERMINANTAL STRUCTURES AND THE CONNECTION TO SUPERSYMMETRY HERMITIAN SUPERMATRICES symmetric supermatrices with respect to the supergroup U(p/q) [ ] Herm (p) [pqg.v.] σ pqg.v. Herm (q) ; σ = σ diagonalization: [ σ1 σ ] [ ] η s1 0 σ = s = σ η σ 2 0 s 2, σ = UsU measure for the eigenvalues: d[σ] Ber p/q (s)d[s]

13 DETERMINANTAL STRUCTURES AND THE CONNECTION TO SUPERSYMMETRY (p q) (1) q {}} { } 1 Ber p/q (s) det s a1 s b2 p } q p s a 1 b2 mixes Cauchy terms with Vandermonde terms (2) Ber p/q (s) det ( ) sa1 ıε p q 1 s b2 ıε s a1 s b2 s a 1 b2 true for arbitrary ε, useful for supersymmetric calculations (1) Basor, Forrester 94 without considering the connection to supersymmetry; (1),(2) Kieburg, Guhr 09 exhibiting the intimate relation to supersymmetry

14 Second example: Special orthogonal group SO (2N)

15 SECOND EXAMPLE: SPECIAL ORTHOGONAL GROUP SO (2N) STARTING POINT Z 2N (κ) = SO(2N) k j=1 det(o κ j2 ) det(o κ j1 ) dµ(o) d µ(o) normalized Haar measure on SO(2N) Interesting for mathematicians: Weyl type character formula Huckleberry, Püttmann, Zirnbauer (2005)

16 SECOND EXAMPLE: SPECIAL ORTHOGONAL GROUP SO (2N) RESULT Z 2N (κ) [ ] 1 Z2N (κ det a1,κ b2 ) Ber k/k ( κ) κ a1 κ b2 We have not used the explicit form of P(E)! Similar results can be found for SO(2N + 1), for special orthogonal groups times reflections, for USp(2N), for their Lie algebras and etc.

17 Aim of our method

18 TWO IMPORTANT QUESTIONS Do all determinantal and Pfaffian structures have the same origin? IF YES: What are the conditions to find such structures?

19 IDEA REASON OF DETERMINANTAL AND PFAFFIAN STRUCTURES The integrand already contains a determinantal structure including the characteristic polynomials! Basor and Forrester: CUE by algebraic rearrangement (1994) Our method considerably extends this idea and makes the connection to supersymmetry!

20 OUR AIM EXAMPLE FOR DETERMINANTAL STRUCTURES (SCHEMATIC) [ ] 1 Z(κa2,κ Z(κ) det b1 ) Ber k/k (κ) κ a2 κ b1 Generating functions for k-point correlations can be expressed by generating functions corresponding to one-point correlation functions! For Pfaffian structures, we find simplifications similar to this formula! The structures carry over to the large N limit!

21 THEOREM FOR DETERMINANTAL STRUCTURES STARTING POINT N Z N (κ,λ) j=1 k 2 (E j κ n2 ) n=1 P(E j ) k 1 (E j κ n1 ) n=1 l 2 m=1 l 1 m=1 (Ej λ m2 ) N (E) 2 d[e] (Ej λ m1 ) RESULT FOR d = N + k 2 k 1 = N + l 2 l 1 0 Z Z d 1 (κ b2,λ a2 ) d (λ b1,λ a2 ) det λ b1 λ a2 Z d (κ a1,κ b2 ) Z d+1 (κ a1,λ b1 ) κ Z N (κ,λ) a1 κ b2 Ber k1 /k 2 (κ) Ber l1 /l 2 (λ)

22 THEOREM FOR PFAFFIAN STRUCTURES STARTING POINT Z N (κ) N g(e 2j 1, E 2j ) j=1 k 2 2N n=1 k 1 j=1 n=1 (E j κ n2 ) 2N (E)d[E] (E j κ n1 ) RESULT FOR AN INTEGER d = N + (k 2 k 1 )/2 0 Z d (κ b1,κ a2 ) (κ b2 κ a2 )Z d 1 (κ a2,κ b2 ) Pf κ b1 κ a2 Z d (κ a1,κ b2 ) (κ b1 κ a1 )Z d+1 (κ a1,κ b1 ) κ Z N (κ) b2 κ a1 Ber k1 /k 2 (κ)

23 GENERAL PROCEDURE (1) algebraic rearrangement (2) determinantal (supersymmetric) structure (3) integration theorems (4) Leibniz expansion (5) identification of the kernels

24 General remarks and conclusions

25 GENERAL REMARKS AND CONCLUSIONS FOR PFAFFIAN AND DETERMINANTAL STRUCTURES structures result from an algebraic construction the joint probability density has only to factorize and the integrals have to be finite, no other requirements applicable to a broad class of ensembles

26 GENERAL REMARKS AND CONCLUSIONS SOME MATRIX ENSEMBLES YIELDING DETERMINANTS

27 GENERAL REMARKS AND CONCLUSIONS SOME MATRIX ENSEMBLES YIELDING PFAFFIAN STRUCTURES

28 GENERAL REMARKS AND CONCLUSIONS FOR MANY CHARACTERISTIC FUNCTIONS IN THE DENOMINATOR determinantal structures for k 1 k 2 N and l 1 l 2 N 0: 1 Z N (κ,λ) Ber k1 /k 2 (κ) Ber l1 /l 2 (λ) λ b1 λ a2 det 0 0 λ a 1 b1 1 κ b 1 κ a1 κ a1 Z 2 (κ a1,λ b1 ) b2 similar results are true for Pfaffian structures two regimes depending on the number of characteristic polynomials

29 GENERAL REMARKS AND CONCLUSIONS RELATION TO ORTHOGONAL POLYNOMIAL METHOD supersymmetric structures are the ultimate reason for the Dyson-Mehta-Mahoux integration theorem determinantal and Pfaffian structures lead to orthogonality and skew-orthogonality relations orthogonal and skew-orthogonal polynomials and their Cauchy-transforms have simple expressions as matrix averages (Akemann, Kieburg, Phillips; in progress) FOR PFAFFIAN STRUCTURES Real and quaternionic ensembles are not distinguishable because their Pfaffian structures have the same origin!

30 GENERAL REMARKS AND CONCLUSIONS OTHER APPLICATIONS ensembles in the presence of an external field and intermediate ensembles all Efetov Wegner terms for the supersymmetric Itzykson Zuber integral supersymmetry calculations in general

31 GENERAL REMARKS AND CONCLUSIONS OTHER APPLICATIONS We use structures of supersymmetry without ever mapping our integrals onto superspace! Supersymmetry without supersymmetry

32 THANK YOU FOR YOUR ATTENTION! M. Kieburg and T. Guhr. Derivation of determinantal structures for random matrix ensembles in a new way J. Phys. A (2010) preprint: arxiv: M. Kieburg and T. Guhr. A new approach to derive Pfaffian structures for random matrix ensembles accepted for publication in J. Phys. A (2010) preprint: arxiv: