Duality, Statistical Mechanics and Random Matrices. Bielefeld Lectures

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1 Duality, Statistical Mechanics and Random Matrices Bielefeld Lectures Tom Spencer Institute for Advanced Study Princeton, NJ August 16, 2016

2 Overview Statistical mechanics motivated by Random Matrix theory Topics: Mean Field models, Phase transitions, Symmetry breaking, Goldstone modes, Infra-red bounds, Mermin-Wagner, Lee-Yang Zeros, SUSY Hyperbolic Sigma model Duality : SUSY Statistical mechanics provides equivalent way to study spectral properties of such disordered quantum systems. Especially useful for analysis of Random Band Matrices.

3 Advantage of Statistical Mechanics Perspective It enables one to get estimates and insights not available by direct analysis. Role of Symmetry and Geometry - transparent. We will see that ordered and disordered phases of lattice spin systems correspond to different spectral types and quantum time evolutions. Wigner Dyson Universality of level spacing can be understood via the study of the saddle manifold.

4 Disadvantages SUSY statistical mechanics is complicated: Spins are 4 by 4 matrices with both commuting and Grassmann variables. Complex measure with oscillations and determinants. For this reason SUSY is often difficult to control analytically. Focus on simple models to illustrate the main ideas

5 Duality and Phase transitions in D 2 Ising model: Disordered T > T c, ordered < T c. Symmetry Z 2. In 2D dual to Grassmann Free Field - Determinant of Dirac. 2D XY dual to 2D Coulomb Gas : plasma phase and dipole phase. Vortices charges. U(1) symmetry. Reinforced Random Walks in 3D strong reinforcement walk is recurrent; weak reinforcement it is transient Dual to SUSY hyperbolic sigma model. Quantum Heisenberg Dual to Random adjacent permutations macroscopic loops - BE condensate, SU(2) symmetry. Anderson Transition in 3D: Localization to delocalization. Dual to SUSY mechanics with U(1, 1 2) Symmetry.

6 Background and History E. Wigner (1955) studied the energy level statistics of a large nucleus. Wigner s brilliant insight: Statistics Energy levels are given by the eigenvalues of a large random matrix - Mean Field model - each matrix element has equal variance. Conjecture: After rescaling, statistical properties of local energy level spacings depend only on symmetry. Universality of Wigner-Dyson Statistics

7 Proof of Universality for Mean Field Models F. J. Dyson, M. Gaudin and M. L. Mehta P. Deift et al; L. Pastur and M. Shcherbina: Invariant ensembles K. Johansson for perturbation of GUE ensembles More Recent work: L. Erdős, B. Schlein and H-T Yau; T. Tao and V. Vu L. Erdős, A. Knowles, H-T Yau, J.Yin (Random Graphs.) Universality of eigenvalue statistics at spectral edges for RBM, W N 5/6, Tracy-Widom (A. Sodin)

8 Gaussian Random Band Matrices: A. GUE: Gaussian Unitary Ensemble, H = H, H ij = 0 H ij H i j = 1 N δ ij δ ji i, j [1, 2,...N] B. RBM: Random Band matrices, H = H, Λ = Large Box H ij = 0, H ij H i j = J ij δ ij δ ji i, j Λ Z d where J ij 0 for i j > W, W = band width. Example: J jk = ( W 2 + 1) 1 (j, k) (2W ) 1 e j k /W. Spectrum H [ 2, 2].

9 Conjectures Beyond Mean Field Theory Fix W 1 and let Λ Z d, Energy in the Bulk Conjecture 1: For D 3 local eigenvalue statistics of Random Band Matrix in large box match those of GUE after scaling. Random Schrödinger version: + λv, v real, random potential, λ 1, local statistics GOE. ID version: If W 2 N = Λ local statistics is GUE. Conjecture 2: In 2D, Localization length e W 2. If N e W 2. Poisson statistics.

10 N! = 0 Simple Examples of Duality e t t N dt = N N+1 e N[s ln s] ds, N N e N 2πN expand about s = 1 0 (t = Ns) P(N) Number of Partitions of N, Hardy - Ramanujan (1918), P(N) = 1 1 2πi m (1 zm ) z N 1 dz 1 4 3N eπ 2N/3 Circle Method in Number Theory.

11 Exercises: Work out corrections to Stirling s formula Why can P(N) be represented as above contour integral? Stationary Phase: Evaluate for large real k > 0: 1 e i k cos(θ) dθ (2πk) 1/2 cos(k π/4) 2π

12 Key Gaussian Integral Identity Let z = (z 1, z 2,..., z N ), z j = x j + iy j x j, y j R If H = H is an N N matrix. det(h E iɛ) 1 = e i z (H E ɛ)z D N z E ɛ = E + i ɛ, ɛ > 0, D N z N j dx j dy j iπ If H is Gaussian of mean 0 then we have e i z Hz H = e 1/2 (z Hz) 2 H = e (z z) 2 /2N GUE

13 GUE Average of det(h E ɛ ) 1 det(h E ɛ ) 1 GUE = e (z z) 2 /2N ie ɛz z D N z e (z z) 2 /2N = 1 2πN e N s2 /2 e i s z z ds Now we can integrate quadratic expression in z z to get 2πN det(h E ɛ ) 1 GUE = Saddle Point: s = E/2 i 1 (E/2) 2 Exercise: Find the asymptotics for E 1.8 e Ns2 /2 (s E ɛ ) N ds

14 Mean field Heisenberg Model Heisenberg spins: s j S 2, j = 1, 2,...N. Magnetic field h R 3, Dual variable x R 3, Z N (β, h) s j S 2 β Temp 1 exp { β N 2N ( s j ) 2 + h s j } Decoupling trick (Hubbard-Stratonovich): exp β N 2N {( s j ) 2 } = ( N 2πβ )3/2 j j N dµ(s j ) e x ( N s j ) e N 2β x2 d 3 x Let F ( x ) = e x s j dµ(s j ) = x 1 sinh x j

15 Z N (β, h) = ( N 2πβ )3/2 Phase transition e N{ln F ( x+h x 2 /2β} d 3 x. When h = 0, Minima of {ln F ( x ) x 2 /2β}: If β 3, x (β) = 0, M(β) = 0 If β > 3, x (β) = r(β)u, r(β) > 0, U S 2, Saddle manifold Second order transition: M(β) = r(β)/β and vanishes continuously as β β c = 3. M 2 (β) 1 N N 2 [ s j ] 2 (β, 0) j

16 Exercises: 1) Prove that for β < 3, {ln F ( x ) x 2 /2β} is a concave function of r = x, F (r) = sinh r r, thus r=0 only saddle point. 2) For GUE 2πN det(h E ɛ ) 1 GUE = Saddle Point: s = E/2 i 1 (E/2) 2 e Ns2 /2 (s E ɛ ) N ds Show that this saddle is correct and analyze the integral by contour deformation. for E 1.8

17 To see the dependence on h expand about the minimum: ln F ( x + h/n ) ln F (r) = 1 N M(β)U h Thus as N, Z N (β, h/n) Z N (β, 0) = e M(β)h U dµ(u) = Mh 1 sinh(m h ) S 2 Formula is valid if we replace s j S 2 by a potential distribution V (s j ) = ((s j ) 2 1) 2, s R 3. The large N limit depends only on the symmetry group and on M(β) - Analogous, to Wigner-Dyson universality!

18 Lattice Spin systems and Lee-Yang Let Λ Z d. For j Λ we consider the spin S j = ±1 Ising; S j = (cos(θ j ), sin(θ j )) XY model; S j S 2 Heisenberg. The partition function Z Λ (β, h) = exp[ βj j,k S j S k + j,k j h S j ] j Λ dµ(s j ) Lee-Yang Theorem If βj j,k 0, then all Zeros of Z Λ (β, h) are on the imaginary h axis. Corollary: f (β, h) = lim Λ Z d Λ 1 log Z Λ (β, h) is analytic if Re h > 0. No Phase transition for Re h > 0. d 2 dh 2 f (β, h) = j [ S 0 S j (β, h) S 0 (β, h) S j (β, h)] <

19 Let Lattice Spin systems and Mean Field For Heisenberg spins: J jk = ( W 2 + 1) 1 (j, k) e j k /W, If β < 3, h = 0. The spin correlations decay exponentially fast for all W. S 0 S j (β, h = 0) e j /l(β) After duality the Gibbs weight is log concave. In 3D, if β > 3 and W is large then there is a magnetization M(β) > 0 and order S 0 S j (β, h = 0) M 2 (β) > 0 as j

20 Mean Field Conjecture for d 3 Conjecture: Let Λ be a periodic cube of side L Z Λ (β, h/ Λ ) = e Mu h dµ(u) (1 + cβ 1 L (d 2) ) Z Λ (β, 0) u =1 Theorem (Fröhlich-Sp) Holds for almost all β for X-Y model. Conjecture Same result for 3D SUSY models: Saddle manifold is U(1, 1 2)/U(1 1) U(1 1). Gives Wigner-Dyson Universality with corrections for 3D RBM.

21 Product of Characteristic Polynomials - GUE Let H denote and N N GUE matrix Define D N (E, δ) = det(h E + δ/n) det(h E δ/n) If E < 2 then as N D N (E, δ) GUE D N (E, 0) GUE = sin( ρ(e)δ)/( ρ(e)δ) Here ρ(e) is proportional to the density of states.

22 Characteristic Polynomial for Gaussian Band Matrices Let H be an N N Gaussian Band matrix of width W. Theorem (T. Shcherbina) In 1D if E < 2, and W 2 N then D N (E, δ) RBM D N (E, 0) RBM = sin( ρ(e)δ)/( ρ(e)δ) Theorem (M. and T. Shcherbina ) If W 2 N then the limit is 1. We shall see that 1D band is similar to a 1D Heisenberg spin chain of length N at with β W 2. The spins in the chain are aligned when W 2 β N.

23 Grassmann Integral Identity Let ψ j, ψ j anti-commuting: ψ k ψ j = ψ j ψ k, ψ k ψ j = ψ j ψ k, (ψ k )2 = (ψ j ) 2 = 0 Ψ = (ψ 1, ψ 2,..., ψ N ) and Ψ = (ψ1, ψ2,..., ψn ) det(h E ɛ ) = e Ψ (H E ɛ)ψ D N Ψ where D N Ψ = n j dψ j dψ j Integration Rule: N j ψ j ψ j D N Ψ 1 Integrals of lower order polynomials vanish.

24 Sketch of proof for GUE If ψ j, ψ j, χ j, χ j, 1 j N, are Grassmann variables then D N (E, δ) = e { ψ (H E)ψ+ χ (H E)χ+ δ N ( ψ ψ χ χ)} Dψ Dχ D N (E, δ) GUE = e {tr Q2 /2N+tr QE+ δ N ( ψ ψ χ χ)} Dψ Dχ ( ) ψ ψ ψ χ where Q = χ ψ χ χ Exercise: Prove this Formula

25 Let X be a 2 2 Hermitian matrix and let DX be the flat measure. e tr Q2 /2N = C N e i tr XQ e N tr X 2 /2 DX the Grassmann s are now quadratic and we can trace them out: D N (E, δ) GUE = C N det(i X Ẽ)N e N tr X 2 /2 DX where Ẽ = diag(e + δ/n, E δ/n) = E I 2 + δ/n σ 3

26 By diagonalizing X we find the saddle X s (at δ = 0) has the following form : ˇX s = {ie/2 ± ρ(e) } I 2 or ˆX s = U {ie/2 I 2 + ρ(e) σ 3 } U where σ 3 = diag(1, 1), U SU(2) and ρ(e) = 2πρ(E) = 1 (E/2) 2. Since det(ix E + δ/nσ 3 ) det(ix E) = e tr(ix E) 1 δσ 3 /N Integration over the saddle manifold ˆX c is e iδ tr (X ie) 1 σ 3 dµ(u) = e iδ ρ tr U σ 3 Uσ 3 dµ(u) = sin( ρ(e)δ)/( ρ(e)δ)

27 ˆX s is Dominant Saddle Manifold The contribution of ˇX s and ˆX s has the same modulus. However at ˇX s, the Jacobian in DX = (λ 1 λ 2 ) 2 dλ 1 dλ 2 dµ(u) vanishes since the eigenvalues λ 1, λ 2 coincide.

28 Spectral information via Green s Function The Green s function for H provides spectral information near E: Let N or box Λ be fixed. Let ε > 0 tr Im (H E iε) 1 ε = tr (H E) 2 + ε 2 π tr δ ɛ(h E) Counts eigenvalues in ɛ - neighborhood of E. tr (H E iε) 1 = d de det(h E iɛ) det(h E iɛ) E=E

29 GUE Average Green s Function ρ N (E, ε) Im 1 N tr(h E iɛ) 1 GUE = Im s 1 SUSY N 2π s 1 e N(s2 1 +s2 2 )/2 (i s 2 E ɛ ) N (s 1 E ɛ ) N R(s 1, s 2 )ds 1 ds 2 where R 1 (s 1 E ɛ ) 1 (i s 2 E ɛ ) 1 Note 1 SUSY = 1. Deform the contour of integration. Analysis about saddle point: s 1 = E/2 i 1 (E/2) 2 There is another sub-dominant saddle point at s 1.

30 Block GUE Matrix Let H 1 and H 2 be independent N N GUE matrices. ( ) H1 c I H N c I N H 2 where c > 0. ρ Λ (E, ε) = 1 2πN Im tr(h E iɛ) 1 GUE Integral in 4 variables. Saddle points solve cubic equation. Universality: local eigenvalue statistics and correlations should be the same as for GUE.

31 Density of States ρ for RBM Let S = (S 1 (j), S 2 (j)) R 2, j Λ Z d, ρ Λ (E, ε) 1 Λ tr(h E iɛ) 1 RBM = S 1 (0) SUSY = C N S 1 (0)e j [W2 ( S)(j) 2 +S(j) 2 ]/2 R j (i S 2 (j) E ɛ ) (S 1 (j) E ɛ ) ds j W = Band Width fixed but Λ Z d - Statistical Mechanics R = det{ W δ ij (S 1 (j) E ɛ ) 1 (i S 2 (j) E ɛ ) 1 }

32 Theorem (Disertori, Pinson, Sp) For RBM with W large and fixed, and E 1.8, ρ Λ (E, ε) is smooth and uniformly bounded as ε 0, Λ Z 3. Remarks: Constantinescu, Felder, Gawedzki and Kupiainen Earlier work for N-orbital model. Theorem gives sharp estimates on density of states for narrow energy windows: ε 1/ Λ. Perturbative methods restricted to ε 1/W.