sources of program support: MRSEC Toeplitz at Brandeis April

Landau Quasiparticle Methods for Calculating Eigenvalues and Eigenvectors of Large Matrices Hui Dai, Zachary Geary, and Leo Kadanoff # # e-mail LeoP@UChicago.edu Matrices are used to describe the modes of oscillation of physical objects. Generally, N by N matrices describe objects with N modes. We often calculate the eigenvectors of large matrices of this kind by making use of calculations for infinite matrices. This approach follows the spirit of Landau quasiparticle methods and also the kinds of scalingt used in critical phenomena. Propagation of information over the whole range of spatial indices in these eigenvectors is demonstrated by power law and logarithmic N-dependence of the eigenvalues and eigenvectors. The dependences can be interpreted in terms of the transfer of information along the line of objects described by the matrix. sources of program support: MRSEC Toeplitz at Brandeis April 2009 1

Toeplitz at Brandeis April 2009 2

Q: How do I find a worthwhile problem? Toeplitz at Brandeis April 2009 2

Q: How do I find a worthwhile problem? A: Find a subject which will have a lasting impact Toeplitz at Brandeis April 2009 2

Q: How do I find a worthwhile problem? A: Find a subject which will have a lasting impact Q: How can I find one of those? Toeplitz at Brandeis April 2009 2

Q: How do I find a worthwhile problem? A: Find a subject which will have a lasting impact Q: How can I find one of those? A: Find something University simple of Chicago MRSEC, and 5-29-08 general which nobody understands Toeplitz at Brandeis April 2009 2

what is a Toeplitz matrix? answer: T(j,k)=T(j-k) When does it arise? Toeplitz at Brandeis April 2009 3

Applications of Toeplitz matrices: Oscillation frequencies of molecules in a solid: You can calculate frequencies of oscillations of a regular solid by Fourier transformation A row of N different molecules have frequencies given by the eigenvalues of an N by N Toeplitz matrix A row of N different molecules have frequencies given by the eigenvalues of an N by N Toeplitz matrix Toeplitz at Brandeis April 2009 4

Other applications 2d Ising model: Onsager solution: N by M system Z=det Q N Q is an M by M matrix correlation functions obtained by local changes in Q correlation functions are given as determinants of Toeplitz matrices. Quantum Information in a two-part system: Mutual information is a determinant of Toeplitz matrix average correlations of eigenvalues in ensemble of random matrices: reduced to determinant of Toeplitz matrix Known application of eigenvalues or eigenfunctions: none yet Toeplitz at Brandeis April 2009 5

Toeplitz eigenvalues: formal properties N 1 N 1 k =0 j =0 T(j k)ψ s k (ψ s k ) tr ψ t k k s = ε s ψ s k (ψ s j ) tr T(j k) = ε s (ψ s k ) tr orthonormality = δ s,t (ψ s j ) tr ψ s k = δ j,k if T(j-k) is real. then We arrange eigenstates in order according to the phase of the eigenvalue completeness tr=transpose j -->N-j-1 this transpose symmetry works like a complex conjugation for hermetian matrices {for all Toeplitz matrices) (ψ s j )* = ψ N s 1 j in this case, the transpose symmetry is hermetian conjugation Toeplitz at Brandeis April 2009 6

describe a family of Toeplitz matrices via Fourier Transform: The relation between a and T is a kind of Fourier transform. a is called the symbol. The inverse Fourier transform. is of the form, which mostly makes sense when z is on the unit circle. a(z) = z j j = T(j) We often write z = e ip note that the function a(z) on unit circle describes family of matrices for values of all N Toeplitz at Brandeis April 2009 7

There is a special case of Toeplitz matrix which was used for statistical mechanics, especially critical properties of 2d Ising Model by University McCoy of Chicago MRSEC, and 5-29-08 Wu Kadanoff Kadanoff-Ceva special case: Hartwig & Fisher Toeplitz at Brandeis April 2009 8

A Singular Case: The matrices T are singular when the function a is singular on the unit circle. We use the form of Fisher and Hartwig for the symbol a α,β (z)=(2-z-1/z) α (-z) β University of Chicago MRSEC, another 5-29-08 IRG way X of writing: a(e -ip ) = (2(1- cos p)) α (-e - ip ) β This form gives for large j-k, β describes jumps in a; α describes zeros T(j-k)~ j-k-β -(1+2 α) Toeplitz at Brandeis April 2009 9

Previous Results Szego looked at det T, which is equivalent to discussing the geometric average of eigenvalues. He proved that, for non-singular a, the large N average has the form (log det T)/N ---> const+ O(1/N) Hartwig and Fisher studied the singular case and found an additional term in the geometric average proportional to (logn)(α 2 β 2 ) / N Harold Widom studied the eigenvalues for large N and proved that in the singular case the distribution of eigenvalues approached the (-infinity,infinity) distribution Lee, Dai, and Bettleheim showed that these eigenvalues each have a correction of the form (logn)f (α,β,z) / N + O(1/ N) Toeplitz at Brandeis April 2009 10

Geometric Strategy Real Problem: on [0, 1,2,...,N-1] Strategy Zero: solve on [-infinity, infinity] (used by previous authors) Strategy One: solve University of Chicago on [0, MRSEC, 1,2, 5-29-08... infinity] (this work) Strategy Two: solve on [-infinity,... N-2, N-1] (future work) Strategy Three: patch together One and Two (future work) Toeplitz at Brandeis April 2009 11

Strategy zero replace j,k = {0,1..., N-1} by j,k in set {-infinity, infinity}. Then eigenvalue problem is k = T(j k)ψ k = εψ j This can be solved by simple Fourier transformation. Or simply try a solution of an exponential University form of Chicago MRSEC, 5-29-08 ψ j = e ipj p has to be real ε = a(e ip ) Then the eigenvalue is Thus, eigenvalue is determined in (-infinity, infinity) case Following Landau-quasiparticle methods we extend this analysis to include finite-n case Toeplitz at Brandeis April 2009 12

ε = a(e ip ) p has to be real [-, ] example eigenvalues: for z =exp(ip) a(z)=(2-z-1/z) α (-z) β β = 1/ 2 α =1/ 3 Toeplitz at Brandeis April 2009 13

As N--> infinity, finite N eigenvalues converge to (-infinity,infinity) result Eigenvalues for N=40, 100, 200, infinity Calculation performed using Mathematica. Eigenvalues for N=101, 201, 401, 801,1601,infinity * means p is close to π/2 We shall particularly study eigenvectors for these eigenvalues Toeplitz at Brandeis April 2009 14

But first jump ahead to results for eigenvalues: j ε (-infinity, infinity) j ε(0, infinity) 1<β<0 case I: all numbers inside the curve are eigenvalues j ε(0, infinity) 0<β<1 case II: no eigenvalues at all j ε{0, N-1} nonetheless we can use the (0,infinity problem) to get information of the eigenfunctions of the [0,n-1] problem Toeplitz at Brandeis April 2009 15

Example I: Positive Translation : T a(z)=z represent an arbitrary state by eigenvalue equation is T Ψ=εΨ so that ɛψ 0 = 0 ψ j z j+1 = ɛ j=0 j=0 ψ j z j ψ(z) = or j=0 ψ j z j ɛψ j = ψ j 1 so that ψ j+1 = ψ j /ɛ for j=0,1,2 all components of the eigenfunction are zero, so no eigenvalue solution Toeplitz at Brandeis April 2009 16 13

Example II: Negative Translation : T a(z) = 1/z ψ(z) = ψ 0 1 + j=1 zψ(z) = ψ j z j 1 j=1 eigenvalue equation is T Ψ=εΨ ψ j z j (1/z) z j = z j-1 for j>0 definition (1/z) 1 = 0 or Text so that ψ j+1 = ɛψ j so that for j=0,1,2,... ψ j = ψ 0 ɛ j for j=0,1,2 require ε to be on or inside unit circle Toeplitz at Berkeley January 9, 2009 17 13

Finite N Results for the eigenvectors Hui Dai, Zachary Geary, Leo Kadanoff The eigenvectors are calculated with the aid of Mathematica Recall: infinite N eigenvectors are of form exp(ipj) with p being real. Hence the magnitude does not vary with j. Large N eigenvectors are roughly of this same form but p has an imaginary part so that the magnitude of the eigenvectors can vary as j varies. In fact, the result looks like Toeplitz at Brandeis April 2009 18

Magnitude of eigenvector finite-n eigenvector roughly varies as exp(ipj) over much of range but p has an imaginary part Empirically Im(p)~ (1+2α) ln N /N drop in eigenfunction over entire range proportional to N -(1+2 α) β = 1/ 2 α =1/ 3 N=101 Toeplitz at Brandeis April 2009 19

Phase of eigenvector increases by approximately (ln i)/i=π/2 as j increases by one unit β = 1/ 2 α =1/ 3 We see that eigenvector is approximately exp(ipj) except at beginning and end Toeplitz at Brandeis April 2009 20

Compare two different N s β = 1/ 2 α =1/ 3 Shape of eigenstate depends only on j/(n-1) except at very beginning and very end. Scale of Eigenstate varies as Toeplitz at Brandeis April 2009 21

Scaling Behavior Three Different Regions of Scaling Central Region: eigenfunction depends upon j/(n-1). shape of wave function versus j/(n-1) is independent of N. Four branches visible near j=n-1. beginning points: near j=0. eigenfunction depends upon j. Size of region remains constant as N goes to infinity. ending points: here near j=n-1. eigenfunction depends upon N-j. Size of region remains constant as N goes to infinity. (I think) Toeplitz at Brandeis April 2009 22

Eigenfunction shows N-independent behavior for small j Toeplitz at Brandeis April 2009 23

For larger j, eigenfunction shows an N- independent behavior when it is plotted as a function of j/(n-1) Toeplitz at Brandeis April 2009 24

Next calculate form of eigenvector: Strategy one Instead of considering j,k in {0,1..., N-1} consider j,k in set {0, 1,...,infinity}. Calculate solution in this case by using Weiner-Hopf (complex analytic) technique. existence of solution is determined by winding number = the (signed) number of times a(z) circles the origin when z circles it once in positive sense Case I: Winding number = -1. There is a unique nontrivial eigenfunction. (example a(z)=1/z) Case II: Winding number > -1. There is no nontrivial eigenfunction. (example a(z)=z, winding number = 1) acceptable solution if 0 < α < β < 1 winding number is 1 when 0 < α < β < 1 case I thus no solution Toeplitz at Brandeis April 2009 25

simplified examples: Case I: many eigenvalues T(j k) = δ j k, 1 N 1 k =0 T(j k)ψ k = εψ j a(z)=1/z note that T transfers information to larger index values ψ j +1 = εψ j j=0,1,2...infinity ψ j = Aε j All eigenvalues on or in unit circle are acceptable ψ j = e ipj Im p greater than or equal to zero, therefore, magnitude of ε=1/z=e -ip eigenfunction decreases as j gets larger. Toeplitz at Brandeis April 2009 26

Case II: no solutions T(j k) = δ j k,1 N 1 k =0 T(j k)ψ k εψ 0 = 0 εψ j = ψ j 1 = εψ j a(z)=z j=0,1,2...infinity j=1,2...infinity note that in this case T transfers information to smaller index values ψ j = 0 only trivial solution: eigenfunction is zero Toeplitz at Brandeis April 2009 27

The Calculational method As said earlier, we can calculate the eigenfunction for the case involving the Hartwig-Fisher form of a(z) in the situation in which the spatial indices run from zero to infinity and the values University of Chicago of MRSEC, the 5-29-08 parameter are in the right range. In the following, I copy the presentation of our joint work provided by my coauthor, Hui Dai. Toeplitz at Brandeis April 2009 28

Equation can be solved by Fourier transformation Toeplitz at Brandeis April 2009 29

positive and negative powers of z non-negative powers of z negative powers of z Note one equation in two unknowns--solve by using analytic properties Toeplitz at Brandeis April 2009 30

first factorize K(z) = K + (z) /[zk (z)] Kψ + = ψ then separate K + (z)ψ + (z) = zk (z)ψ (z) then solve K + (z)ψ + (z) = zk (z)ψ (z) ψ + = Const / K + = j=0 z j ψ j = const Toeplitz at Brandeis April 2009 31

G + path surrounds all singularities outside unit circle Toeplitz at Brandeis April 2009 32

Pushing the Contour Outward z Start with zero in a(z )-ε (at z =zc) (gives branch cut in exponent) branch cut in a(z ) blue contour at unit circle pole at z =z Convert to branch point, cut, in a(z )-ε branch cut in a(z ) circle at infinity Toeplitz at Brandeis April 2009 33

Result: An evaluation of the eigenfunction terms: = z j ψ j j (z / z c ) j j =0 z c = i e iδ /N e γ(logn )/N a few low powers of z give some N-independent terms for small j term in integrand like (t-1) 2α gives additive term ψ j 1/ j 2α+1 Toeplitz at Brandeis April 2009 34

Weiner Hopf solution Exact Eigenfunction β = 1/ 2 α =1/ 3 Toeplitz at Brandeis April 2009 35

Weiner Hopf solution Exact Eigenfunction β = 1/ 2 α =1/ 3 looks the same except for j=100 point Toeplitz at Brandeis April 2009 35

Weiner-Hopf gives excellent solution for j/n<0.9 Plot absolute error in WH approximation vs j/(n-1) Plot relative error in WH approximation vs j/(n-1) Here N=1000, error is of order 10-7 Relative error remains small until large values of j/(n-1) Toeplitz at Brandeis April 2009 36

Eigenfunction is a sum of an exponential and an algebraic term N=100 N=1000 red = (zc) -j green=.2/(j+1) 1+2 α blue=data Toeplitz at Brandeis April 2009 37

terms: (z / z c ) j j =0 explains overall exponential decay, and period approximately 4, z = e iq e γ (log N )/N c a few low powers of z give some N-independent University of Chicago terms MRSEC, for 5-29-08 small j explains N-independent bumps for small j term in integrand like (t-1) 2α gives additive term ψ j 1/ j 2α+1 explains behavior of eigenvalue... Toeplitz at Brandeis April 2009 38

Explain behavior of eigenvalue Near j=n-1 end two terms balance out and permit system to reach end point. Exponential decay term ψ N 1 z c N = e γ(logn )/N N γ logn = e = N γ balances against algebraic decay ψ N 1 N (2α+1) giving γ = 2α +1 p = p 0 + i(2α +1)(logN) / N + O(1/ N) and an eigenvalue of the form and a p-value of the form ε = (2 e ip 0 e ip 0 ) α ( e ip 0 ) β [1 i logn N (2α +1)(β + α cot(p 0 / 2))] This is the first determination of the eigenvalue shift for finite N Toeplitz at Brandeis April 2009 39

β=-1/2 α=1/4 theory gives 1+2α=1.5 observation gives 1.564 offset=exp(-1.16)=.22 β=-1/2 α=1/3 theory gives 1+2α=1.67 observation gives 1.749 offset=exp(-1.7) =.18 Toeplitz at Brandeis April 2009 40

Conclusions I: What do we have? We have an accurate method of calculating eigenfunctions which works except at the low end of the eigenfunction (j near N-1 for negative β). The method works for both sign of β since the interchange of β and -β simply trades which region of j is the high side. no exponential term for positive β We have a determination of eigenvalue good to (log N)/N. These ideas apparently work across the entire eigenvalue spectrum but have not been fully evaluated near the ends of the spectrum. We do NOT have an understanding of the Hartwig-Fisher result for the determinant. Maybe the result comes from the ends of the spectrum. We expect more progress soon. Toeplitz at Brandeis April 2009 41

j ε (-infinity, infinity) Conclusions II: Eigenvalues j ε(0, infinity) 1<β<0 j ε(0, infinity) 0<β<1 j ε{0, N-1} Toeplitz at Brandeis April 2009 42

Conclusions III: information transfer j ε (-infinity, infinity) j ε(0, infinity) 1<β<0 K must transfer information in both directions information mostly transferred from zero to infinity j ε(0, infinity) 0<β<1 information mostly transferred from infinity to zero j ε{0, N-1} K must transfer information in both directions Toeplitz at Brandeis April 2009 43 39

References Hui Dai, Zachary Geary, and Leo Kadanoff this work, arxiv:0901.3436 (2009) Toeplitz at Brandeis April 2009 44