Spin matrix elements in 2D Ising model on the finite lattice

Spin matrix elemen ts in 2D Ising mo del on the finite latt ice A. I. Bugrij ∗ , O. Liso vyy ∗ , † ∗ Bogolyub o v Institute for Theoretical Physics Metrolohic hna str., 14-b, K yiv-143 , 03143, Ukraine † Lab oratoire de Math ´ ematiques et Ph ysique Th´ eo rique CNRS /UMR 6083, Univ ersit ´ e de T ours, P arc de Gr andmon t, 37200 T ours, F rance Abstract W e present explicit form ulas for all spin matrix elements in the 2D Ising mo del with the nearest neighbor in teraction on t he finite per io dic square lattice. Thes e expressions generalize the known res ults [1, 3 , 6] (coincide with them in the appropriate limits) and fulfill the test of straightforward transfer matrix calculations for finite N . 1 Eigen v alues and eigen v ectors of the transfer matrix It is w ell-kno wn (see [7, 8 ]) that the sp ectrum of 2 N × 2 N transfer matrix, corresp onding to Ising mo del on the p erio dic squ are lattice, consists of tw o sets: λ = (2 s ) N/ 2 exp  1 2 ( ± γ (0) ± γ ( 2 π / N ) ± . . . ± γ (2 π − 2 π / N ))  , (1) λ = (2 s ) N/ 2 exp  1 2 ( ± γ ( π / N ) ± γ (3 π / N ) ± . . . ± γ (2 π − π / N ))  , (2) where s = sinh 2 K and K is th e Ising coupling constan t. The function γ ( q ) is defin ed as the p ositiv e ro ot of the equation cosh γ ( q ) = s + s − 1 − cos q . whic h is the lattice analog of the relativistic energy disp ersion la w. The n um b er of minuses in (1) is ev en in ferromagnetic ( s > 1) and o dd in paramagnetic (0 < s < 1) ph ase, w hile the n um b er of minuses in (2) is ev en in b oth phases. The eigen v alues (1) (or (2)) corresp ond to eigen vec tors that are o dd (resp. even) un der spin reflection. The notation and terminology , introdu ced in [6] for the analysis of con tinuum limit, are also v ery con v enient on th e lattice. In what f ollo ws, o dd and even eigen vect ors of the Ising transf er matrix will b e interpreted as m ultiparticle states f r om the Ramond and Nev eu-Sc hw artz s ecto r. Quasimomen ta of R–particles can b e equ al to 2 π N j ( j = 0 , 1 , . . . , N − 1), while for NS–particles they take on the v alues 2 π N  j + 1 2  ( j = 0 , 1 , . . . , N − 1). Eac h eigenstate consists of p articles of only one t y p e, and their quasimoment a must b e different. W e will denote by | p 1 , . . . , p K i N S ( R ) the normalized ei genstate, cont aining particles with t he momen ta p 1 , . . . , p K . Sin ce R–secto r in paramagnetic phase con tains t he sta te | 0 i R (one particle with zero momen tum), it w ill b e con v enient to d en ote NS and R v acua by |∅i N S and |∅i R . The goal of the presen t pap er is to fi nd matrix el emen ts N S h p 1 , . . . , p K | σ | q 1 , . . . , q L i R of the Ising spin σ in the describ ed basis of normalized eigenstates. (R–R and NS–NS matrix elemen ts v anish due to Z 2 -symmetry of the mo del). 1 2 Lattice form factors and scaling limit All n -p oin t correlation fu nctions in the Ising mo del on the cylinder an d torus can b e easily expressed via spin matrix elemen ts. Ho wev er, kno wn r esu lts w ere obtained in rather in v erse w a y . A t the first stage, 2-p oint fun ctions are expressed through the determinants of certain T o eplitz matrices with a size that d ep ends on the separation of correlating spins. T o extract the analytic dep endence on the distance from these representati ons, a lot of further w ork w as needed [2]. The fi nal answ er [3, 4] allo ws to calculate squared form factors on the cylinder (on the infin ite lattice the ab o v e program wa s r ealiz ed earlier in [9 , 10 , 12]):   N S h∅| σ | q 1 , . . . , q L i R   2 = ξ ξ T L Y j =1 e − ν ( q j ) N sinh γ ( q j ) Y 1 ≤ i 1). In this case NS (R) momen ta tak e on the v alues π , π / 3, − π / 3 ( 0, 2 π / 3, − 2 π / 3). E ac h state con tains either t w o particles or no particles at all. Since the integrals (4) and (5) can b e alternativ ely wr itten as ξ 4 T = Q q ( R ) Q p ( N S ) sinh 2 γ ( q )+ γ ( p ) 2 Q q ( R ) Q p ( R ) sinh γ ( q )+ γ ( p ) 2 Q q ( N S ) Q p ( N S ) sinh γ ( q )+ γ ( p ) 2 , ν ( q ) = ln Q p ( N S ) sinh γ ( q )+ γ ( p ) 2 Q p ( R ) sinh γ ( q )+ γ ( p ) 2 , then to v erify (12) it suffices to pr o ve ten relations: N S h∅| σ |∅i R 2 = sinh γ 0 + γ π/ 3 2 sinh γ π + γ 2 π/ 3 2 sinh 2 γ π/ 3 + γ 2 π/ 3 2 sinh γ 2 π / 3 sinh γ π / 3 sinh γ 0 + γ 2 π/ 3 2 sinh γ π + γ π/ 3 2 , N S h− π / 3 , π / 3 | σ | 2 π / 3 , − 2 π / 3 i R 2 = sinh γ 0 + γ 2 π/ 3 2 sinh γ π + γ π/ 3 2 sinh 2 γ π/ 3 + γ 2 π/ 3 2 9 sinh γ 2 π / 3 sinh γ π / 3 sinh γ 0 + γ π/ 3 2 sinh γ π + γ 2 π/ 3 2 , 3 N S h∅| σ | 2 π / 3 , − 2 π / 3 i R 2 = sinh γ 0 + γ π/ 3 2 sinh γ 0 + γ 2 π/ 3 2 12 sinh γ 2 π / 3 sinh γ π / 3 sinh γ π + γ π/ 3 2 sinh γ π + γ 2 π/ 3 2 sinh 2 γ π/ 3 + γ 2 π/ 3 2 , N S h− π / 3 , π / 3 | σ |∅i R 2 = sinh γ π + γ π/ 3 2 sinh γ π + γ 2 π/ 3 2 12 sinh γ 2 π / 3 sinh γ π / 3 sinh γ 0 + γ π/ 3 2 sinh γ 0 + γ 2 π/ 3 2 sinh 2 γ π/ 3 + γ 2 π/ 3 2 , N S h∅| σ | 0 , 2 π / 3 i R 2 = N S h∅| σ | 0 , − 2 π/ 3 i R 2 = 1 12 sinh γ 0 + γ π 2 sinh γ 0 + γ π/ 3 2 sinh γ π + γ π/ 3 2 sinh γ π / 3 , N S h− π / 3 , π | σ |∅i R 2 = N S h π / 3 , π | σ |∅i R 2 = 1 12 sinh γ 0 + γ π 2 sinh γ 0 + γ 2 π/ 3 2 sinh γ 2 π / 3 sinh γ π + γ 2 π/ 3 2 , N S h− π / 3 , π / 3 | σ | 0 , 2 π / 3 i R 2 = N S h− π / 3 , π / 3 | σ | 0 , − 2 π / 3 i R 2 = 4 sinh γ 0 + γ π/ 3 2 sinh γ π + γ π/ 3 2 9 sinh γ π / 3 sinh γ 0 + γ π 2 , N S h− π / 3 , π | σ | 2 π / 3 , − 2 π / 3 i R 2 = N S h π / 3 , π | σ | 2 π / 3 , − 2 π / 3 i R 2 = 4 sinh γ 0 + γ 2 π/ 3 2 sinh γ π + γ 2 π/ 3 2 9 sinh γ 2 π / 3 sinh γ 0 + γ π 2 , N S h− π / 3 , π | σ | 0 , 2 π / 3 i R 2 = N S h π / 3 , π | σ | 0 , − 2 π / 3 i R 2 = 1 9 , N S h− π / 3 , π | σ | 0 , − 2 π / 3 i R 2 = N S h π / 3 , π | σ | 0 , 2 π / 3 i R 2 = 4 9 . This ind eed can b e done — with a little b it cumbers ome but straightforw ard calculation 1 . W e ha v e p erformed a similar c hec k for sm all N up to N = 4 and we hav e no doubt in th e v alidit y of (12) for arb itrary N . The rigorous pro of of this form ula will complete, in a sense, the stud y of the 2D Ising mo del in zero field. ———————————– W e thank B. Banos, V. N. R u btso v and V. N. Shadura for help and n umerous stim ulating discussions. Th is work was sup p orted b y the INT AS p rogram un der gran t INT AS-00-000 55. References [1] B. Berg, M. Karo w s ki, P . W eisz , Construction of G r e en ’s f unctions fr om an exact S matrix , Ph ys. Rev. D19 , (1979), 2477–247 9. [2] A. I. Bugrij, Corr elation function of the tw o-dimensional Ising mo del on the finite lat tic e. I , Theor. Math. Phys. 127 , (2001), 528–548 ; hep- th/0011104 . [3] A. I. 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