Toeplitz determinants in Mathematical Physics Alberto Ibort fest, - PowerPoint PPT Presentation

Fernando Falceto Theoretical Physics Department. Universidad de Zaragoza Toeplitz determinants in Mathematical Physics Alberto Ibort fest, ICMAT, Madrid, March 5-9, 2018. In collaboration with: Filiberto Ares Jos´ e G. Esteve Amilcar de Queiroz

Aim of the talk ◮ We review the main steps in our progress towards the understanding of Toeplitz determinants. ◮ We discuss connections of the latter with physics, namely: the Ising model and entanglement entropy of fermionic chains. ◮ We emphasize the impulse that physics has given to the development of the theory. ◮ Finally we present new results and conjectures on the subject. 2 / 32

Aim of the talk ◮ We review the main steps in our progress towards the understanding of Toeplitz determinants. ◮ We discuss connections of the latter with physics, namely: the Ising model and entanglement entropy of fermionic chains. ◮ We emphasize the impulse that physics has given to the development of the theory. ◮ Finally we present new results and conjectures on the subject. 2 / 32

Aim of the talk ◮ We review the main steps in our progress towards the understanding of Toeplitz determinants. ◮ We discuss connections of the latter with physics, namely: the Ising model and entanglement entropy of fermionic chains. ◮ We emphasize the impulse that physics has given to the development of the theory. ◮ Finally we present new results and conjectures on the subject. 2 / 32

Aim of the talk ◮ We review the main steps in our progress towards the understanding of Toeplitz determinants. ◮ We discuss connections of the latter with physics, namely: the Ising model and entanglement entropy of fermionic chains. ◮ We emphasize the impulse that physics has given to the development of the theory. ◮ Finally we present new results and conjectures on the subject. 2 / 32

Aim of the talk ◮ We review the main steps in our progress towards the understanding of Toeplitz determinants. ◮ We discuss connections of the latter with physics, namely: the Ising model and entanglement entropy of fermionic chains. ◮ We emphasize the impulse that physics has given to the development of the theory. ◮ Finally we present new results and conjectures on the subject. Based on: - P. Deift, A. Its, I. Krasovsky, Comm.Pure Appl.Math. 66. arXiv:1207.4990 - F. Ares, J. G. Esteve, F. F., Phys. Rev. A 90, (2014) - F. Ares, J. G. Esteve, F. F., A. R. de Queiroz, J. Stat. Mech. 063104, (2017) - F. Ares, J. G. Esteve, F. F., A. R. de Queiroz, arXiv:1801.07043 , (2018) 2 / 32

Toeplitz matrices (Toeplitz 1907) Symbol f : S 1 → C , f ∈ L 1 � π t k = 1 f ( θ )e − i kθ d θ 2 π − π 3 / 32

Toeplitz matrices (Toeplitz 1907) Symbol f : S 1 → C , f ∈ L 1 � π t k = 1 f ( θ )e − i kθ d θ 2 π − π Toeplitz Matrix with symbol f :   t 0 t − 1 t − 2 · · · · · · · · · · · · t 1 − n . .   t 1 t 0 t − 1 t − 2 .     . ... .   t 2 t 1 t 0 t − 1 .    . .  ... ... ... ... . .   . t 2 .   T n ( f ) = . .  ... ... ... ...  . .   . t − 2 .   .  ...  .  . t 1 t 0 t − 1 t − 2    .   . . t 2 t 1 t 0 t − 1   t n − 1 · · · · · · · · · · · · t 2 t 1 t 0 3 / 32

Toeplitz determinant Introduce the Toeplitz determinant with symbol f D n ( f ) = det T n ( f ) 4 / 32

Toeplitz determinant Introduce the Toeplitz determinant with symbol f D n ( f ) = det T n ( f ) Szeg˝ o theorem (1915): � 1 � π � For f : S 1 → R + continuous and [ f ] = exp log f ( θ )d θ 2 π − π n →∞ ( D n ( f )) 1 /n = [ f ] lim 4 / 32

Toeplitz determinant Introduce the Toeplitz determinant with symbol f D n ( f ) = det T n ( f ) Szeg˝ o theorem (1915): � 1 � π � For f : S 1 → R + continuous and [ f ] = exp log f ( θ )d θ 2 π − π n →∞ ( D n ( f )) 1 /n = [ f ] lim Our cooperation started from a conjecture which I found. It was about a determinant considered by Toeplitz and others, formed with the Fourier-coefficients of a function f (x). I had no proof, but I published the conjecture and the young Szeg˝ o found the proof... G. P´ olya, Mathematische Annalen, 1915 4 / 32

Toeplitz determinant Introduce the Toeplitz determinant with symbol f D n ( f ) = det T n ( f ) Szeg˝ o theorem (1915): � 1 � π � For f : S 1 → R + continuous and [ f ] = exp log f ( θ )d θ 2 π − π n →∞ ( D n ( f )) 1 /n = [ f ] lim In other words D n ( f ) = e o ( n ) [ f ] n 4 / 32

Toeplitz determinant Introduce the Toeplitz determinant with symbol f D n ( f ) = det T n ( f ) Szeg˝ o theorem (1915): � 1 � π � For f : S 1 → R + continuous and [ f ] = exp log f ( θ )d θ 2 π − π n →∞ ( D n ( f )) 1 /n = [ f ] lim In other words D n ( f ) = e o ( n ) [ f ] n Can we say something about o ( n ) ? 4 / 32

Toeplitz determinant Szeg˝ o Strong limit theorem (Szeg˝ o 1952, Johanson 1988): Let f : S 1 → C , with log f ∈ L 1 , call � π s k = 1 log f ( θ )e − i kθ d θ 2 π − π Hence if ∞ � | k || s k | 2 < ∞ k = −∞ D n ( f ) � ∞ k =1 ks k s − k lim = e e ns 0 n →∞ 5 / 32

Toeplitz determinant Szeg˝ o Strong limit theorem (Szeg˝ o 1952, Johanson 1988): Let f : S 1 → C , with log f ∈ L 1 , call � π s k = 1 log f ( θ )e − i kθ d θ 2 π − π Hence if ∞ � | k || s k | 2 < ∞ k = −∞ D n ( f ) � ∞ k =1 ks k s − k lim = e e ns 0 n →∞ ∞ � Comparing with previous slide, o ( n ) = ks k s − k + o (1) k =1 5 / 32

Toeplitz determinant Szeg˝ o Strong limit theorem (Szeg˝ o 1952, Johanson 1988): Let f : S 1 → C , with log f ∈ L 1 , call � π s k = 1 log f ( θ )e − i kθ d θ 2 π − π Hence if ∞ � | k || s k | 2 < ∞ k = −∞ D n ( f ) � ∞ k =1 ks k s − k lim = e e ns 0 n →∞ ∞ � Comparing with previous slide, o ( n ) = ks k s − k + o (1) k =1 ∞ � | k || s k | 2 = ∞ ? What if k = −∞ 5 / 32

Ising model in two dimensions ( σ x,y ) Kaufman and Onsager (1949) � σ 0 , 0 σ n,n � = D n ( f Is ) f Is = e iArg φ , φ ( θ ) = 1 − A e i θ , with A = (sinh k B T ) − 2 . 2 J 6 / 32

Ising model in two dimensions ( σ x,y ) Kaufman and Onsager (1949) � σ 0 , 0 σ n,n � = D n ( f Is ) f Is = e iArg φ , φ ( θ ) = 1 − A e i θ , with A = (sinh k B T ) − 2 . 2 J T < T c T = T c T > T c φ ( θ ) φ ( θ ) φ ( θ ) − π − π − π π π π A < 1 A = 1 A > 1 6 / 32

Ising model, T < T c ( A < 1 ). φ ( θ ) − i log f Is π/ 2 − π π − π/ 2 k = −∞ | k || s k | 2 < ∞ � ∞ For A < 1 , log f Is ∈ C 1+ ǫ ⇒ ⇒ ⇒ Szeg˝ o Strong Limit Theorem applies. 7 / 32

Ising model, T < T c ( A < 1 ). φ ( θ ) − i log f Is π/ 2 − π π − π/ 2 k = −∞ | k || s k | 2 < ∞ � ∞ For A < 1 , log f Is ∈ C 1+ ǫ ⇒ ⇒ ⇒ Szeg˝ o Strong Limit Theorem applies. Hence D n ( f Is ) � ∞ k =1 ks k s − k lim = e e ns 0 n →∞ with ∞ s k = − A | k | ks k s − k = 1 � 4 log(1 − A 2 ) s 0 = 0 , 2 k , k =1 7 / 32

Ising model, T < T c ( A < 1 ). φ ( θ ) − i log f Is π/ 2 − π π − π/ 2 Then n →∞ � σ 0 , 0 σ n,n � = lim lim n →∞ D n ( f Is ) = � ∞ k =1 ks k s − k = (1 − A 2 ) 1 / 4 . = e From which we derive the spontaneous magnetization n →∞ � σ 0 , 0 σ n,n � 1 / 2 = (1 − A 2 ) 1 / 8 M 0 = lim 8 / 32

Ising model, T < T c ( A < 1 ). φ ( θ ) − i log f Is π/ 2 − π π − π/ 2 ...and lo and below I found it. It was a general formula for the evaluation of Toeplitz matrices. The only thing I did not know was how to fill out the holes in the mathematics and show the epsilons and deltas and all that... ...the mathematicians got there first... L. Onsager, 1971. n →∞ � σ 0 , 0 σ n,n � 1 / 2 = (1 − A 2 ) 1 / 8 M 0 = lim 8 / 32

Ising model, T = T c ( A = 1 ). φ ( θ ) − i log f Is π/ 2 − π π − π/ 2 ∞ f Is has jumps, s k = − 1 � | k || s k | 2 = ∞ 2 k ⇒ k = −∞ 9 / 32

Ising model, T = T c ( A = 1 ). φ ( θ ) − i log f Is π/ 2 − π π − π/ 2 ∞ f Is has jumps, s k = − 1 � | k || s k | 2 = ∞ 2 k ⇒ k = −∞ R R � � | e i θ − e i θ r | 2 α r f ( θ ) = e V ( θ ) g βr ( θ − θ r ) , θ, θ r ∈ ( − π, π ] r =1 r =1 g β ( θ ) = e i( θ − π sgn( θ )) β • V ( θ ) periodic and smooth enough, 9 / 32

Ising model, T = T c ( A = 1 ). φ ( θ ) − i log f Is π/ 2 − π π − π/ 2 ∞ f Is has jumps, s k = − 1 � | k || s k | 2 = ∞ 2 k ⇒ k = −∞ R R � � | e i θ − e i θ r | 2 α r f ( θ ) = e V ( θ ) g βr ( θ − θ r ) , θ, θ r ∈ ( − π, π ] r =1 r =1 g β ( θ ) = e i( θ − π sgn( θ )) β • V ( θ ) periodic and smooth enough, • f has zeros and/or jump discontinuities at θ = θ r , r = 1 , . . . , R . 9 / 32

Ising model, T = T c ( A = 1 ). φ ( θ ) − i log f Is π/ 2 − π π − π/ 2 ∞ f Is has jumps, s k = − 1 � | k || s k | 2 = ∞ 2 k ⇒ k = −∞ R R � � | e i θ − e i θ r | 2 α r f ( θ ) = e V ( θ ) g βr ( θ − θ r ) , θ, θ r ∈ ( − π, π ] r =1 r =1 g β ( θ ) = e i( θ − π sgn( θ )) β • V ( θ ) periodic and smooth enough, • f has zeros and/or jump discontinuities at θ = θ r , r = 1 , . . . , R . f Is = g 1 / 2 9 / 32