Large distance asymptotic behavior of the emptiness formation probability of the XXZ spin- 1 2 Heisenberg chain

Transcription

1 LPENSL-TH-0/0 Large dstance asymptotc behavor of the emptness formaton probablty of the XXZ spn- Hesenberg chan N. Ktanne, J. M. Mallet, N. A. Slavnov 3, V. Terras 4 Abstract Usng ts multple ntegral representaton, we compute the large dstance asymptotc behavor of the emptness formaton probablty of the XXZ spn- Hesenberg chan n the massless regme. Graduate School of Mathematcal Scences, Unversty of Tokyo, Japan, ktanne@ms.u-tokyo.ac.jp On leave of absence from Steklov Insttute at St. Petersburg, Russa Laboratore de Physque, UMR 567 du CNRS, ENS Lyon, France, mallet@ens-lyon.fr 3 Steklov Mathematcal Insttute, Moscow, Russa, nslavnov@m.ras.ru 4 Department of Physcs and Astronomy, Rutgers Unversty, USA, vterras@physcs.rutgers.edu On leave of absence from LPMT, UMR 585 du CNRS, Montpeller, France

2 Emptness formaton probablty at large dstance The Hamltonan of the XXZ spn- Hesenberg chan s gven by H = M σ x m σm+ x + σy m σy m+ + σz m σz m+.. m= Here s the ansotropy parameter, σm x,y,z denote the usual Paul matrces actng on the quantum space at ste m of the chan. The emptness formaton probablty τm the probablty to fnd n the ground state a ferromagnetc strng of length m s defned as the followng expectaton value m σk z τm = ψ g ψ g,. k= where ψ g denotes the normalzed ground state. In the thermodynamc t M, ths quantty can be expressed as a multple ntegral wth m ntegratons [,, 3, 4, 5]. Recently, n the artcle [6], a new multple ntegral representaton for τm was obtaned. It leads n a drect way to the known answer at the free fermon pont = 0 [9], n partcular usng a saddle pont method, and to ts frst exact determnaton outsde the free fermon pont, namely at = [0]. The purpose of ths letter s to present the evaluaton of the asymptotc behavor of τm at large dstance m, n the massless regme < <, va the saddle pont method. We fnd log τm m m =log π + R 0 dω ω snh ω ω π cosh,.3 snh πω ω snh cosh ω where cos =,0<<π. If s commensurate wth π n other words f e s a root of unty, then the ntegral n.3 can be taken explctly n terms of ψ-functon logarthmc dervatve of Γ-functon. In partcular for = π and = π 3 respectvely = 0 and = / we obtan from.3 log τm m m = log, =0, log τm m m = 3 log 3 3log, =, whch concdes wth the known results obtaned respectvely n [7, 8, 9] and n [, 0]. For the partcular case of the XXX chan =, = 0 the asymptotc behavor can be evaluated also by the saddle pont method and t s gven by log τm m m =log Γ 3 4 Γ Γ 4.4 log0.599,.5

3 whch s n good agreement wth the known numercal result log0.598, obtaned n []. Below, we explan the man features of our method. A more detaled account of the proofs and technques nvolved wll be publshed later. The saddle pont method The multple ntegral representaton for τm obtaned n [6] can be wrtten n the form τm = sn m m m π D d m λ F {λ},m wth m a>b F {λ},m = snh π λ a λ b snhλ a λ b snhλ a λ b + ξ,...ξ m a>b m snhλa snhλ a + m,. a= cosh π λ a sn det m m.. snhλ snhξ a ξ b j ξ k snhλ j ξ k Here the ntegraton doman D s <λ <λ < <λ m <. Followng the standard arguments of the saddle pont method we estmate the ntegral. by the maxmal value of the ntegrand. Let {λ } be the set of parameters correspondng to ths maxmum. They satsfy the saddle pont equatons and for large m we assume that ther dstrbuton can be descrbed by a densty functon ρλ : ρλ j = m mλ j+.3 λ j. Thus for large m, one can replace sums over the set {λ } by ntegrals. ntegrable on the real axs, then Namely, f fλ s m m m fλ j j= m j= j k fλ j λ j λ k fλρλ dλ, V.P. fλ λ λ ρλ dλ, k m..4 Due to.4 t s easy to see that n the pont λ,...,λ m the products n the second lne of. behave as expc m. 3

4 Our goal s now to estmate the behavor of the term F {λ },m. To do ths we factorze the determnant n. as follows for large m: sn det m snhλ j ξ k snhλ j ξ k wth Indeed, for m one has = π m det m δ jk Kλ j λ k πmρλ k det m Kλ = det m δ jk Kλ j λ k πmρλ k det m snh π λ j ξ k snh π λ j ξ k,.5 sn snhλ snhλ +..6 m =det m snh π λ j ξ k Kλ j λ l πmρλ l snh π λ l ξ k l= det m snh π λ j ξ k Kλ j µ π dµ snh π µ ξ k m sn = det m π snhλ j ξ ksnhλ j ξ. k.7 Here we have used the fact that the functon / snh π λ j ξ solves the Leb ntegral equaton for the densty of the ground state of the XXZ magnet [3] and we have used the notatons of [6]. The second determnant n the r.h.s. of.5 s a Cauchy determnant, hence, F {λ },m= m π m +m m snh π λ a λ b a>b m cosh m π λ a a= det m δ jk Kλ j λ k πmρλ k..8 The behavor of the determnant n.8 can be estmated va Hadamard nequalty det m a jk max a jk m m m..9 appled to the above determnant and to the determnant of the nverse matrx, whch shows that m m log det m δ jk Kλ j λ k πmρλ k =0..0 4

5 The last equaton means that det m δ jk Kλ j λ k /πmρλ k does not contrbute to the leadng term of the asymptotcs. Hence, t can be excluded from our consderatons. Thus, up to subleadng correctons of the exponental type the emptness formaton probablty behaves as π m τm e m S 0, m,. wth S 0 S{λ }= m snh π m log λ a λ b snhλ a λ b snhλ a λ b + + m a= a>b m snhλ a / snhλ a + / log cosh π.. λ a Here the parameters {λ } are the solutons of the saddle pont equatons In our case the system.3 has the form π tanh πλ j = m m k= k j S 0 λ j cothλ j / cothλ j + / =0..3 π coth π λ j λ k cothλ j λ k cothλ j λ k +..4 Usng.4 we transform.4 nto the ntegral equaton for the densty ρλ π tanh πλ cothλ / cothλ + / π = V.P. coth π λ µ cothλ µ cothλ µ + ρµ dµ..5 Respectvely the acton S 0 takes the form snhλ / snhλ + / S 0 = dλρλlog cosh π λ + snh π λ µ dµdλρλρµlog..6 snhλ µ snhλ µ + 5

6 Snce the kernel of the ntegral operator n.5 depends on the dfference of the arguments, ths equaton can be solved va Fourer transform. Then ˆρω = Makng the nverse Fourer transform we fnd e ωλ ρλ dλ = cosh ω cosh ω..7 ρλ = πλ cosh cosh πλ,.8 whch obvously satsfes the needed normalsaton condton for densty ntegral on the real axs equals one. It remans to substtute.7,.8 nto.6, and after straghtforward calculatons we arrve at S 0 = R 0 dω ω snh ω ω π cosh..9 snh πω ω snh cosh ω Thus, we have obtaned.3. In the case of the XXX chan = one should rescale λ j λ j, ξ j ξ j n the orgnal multple ntegral representaton. for τm and then proceed to the t 0. The remanng computatons are then very smlar to the ones descrbed above, therefore we present here only the man results. The behavor of τm snowgvenby The acton S 0 n the saddle pont has the form λ /λ + / S 0 = log cosh ρλ dλ πλ π tanh πλ λ λ τm π m e m S 0, m..0 = V.P. dµdλρλρµlog The analog of the ntegral equaton.5 n the XXX case s π coth πλ µ snh πλ µ λ µ λ µ + λ µ λ µ +.. ρµ dµ.. The soluton of ths equaton s ρλ = πλ cosh..3 coshπλ Substtutng.3 nto. we fnally arrve at.5. 6

7 Acknowledgments N. K. s supported by JSPS grant P077. N. K. would lke to thank M. Jmbo for help. N. S. s supported by the grants RFBR , Foundaton of the Support of Russan Scence, Leadng Scentfc Schools , the Program Nonlnear Dynamcs and Soltons and by CNRS. J.M. M. s supported by CNRS. V. T s supported by DOE grant DE-FG0-96ER40959 and by CNRS. N. K, N. S. and V. T. would lke to thank the Theoretcal Physcs group of the Laboratory of Physcs at ENS Lyon for hosptalty, whch makes ths collaboraton possble. We also would lke to thank the organzors of the 6th Internatonal Workshop Conformal Feld Theory and Integrable Models held n Chernogolovka, September 5-, 00, for the nce and stmulatng scentfc and extra-scentfc atmosphere they succeeded to generate. References [] M. Jmbo, K. Mk, T. Mwa and A. Nakayashk, Phys. Lett. A [] M. Jmbo and T. Mwa, Journ. Phys. A: Math. Gen., [3] M. Jmbo and T. Mwa, Algebrac analyss of solvable lattce models AMS, 995. [4] N. Ktanne, J. M. Mallet and V. Terras, Nucl. Phys. B, 554 [FS] , mathph/ [5] N. Ktanne, J. M. Mallet and V. Terras, Nucl. Phys. B, 567 [FS] ; mathph/ [6] N. Ktanne, J. M. Mallet, N. A. Slavnov, V. Terras, Nucl. Phys. B 64 [FS] ; hep-th/ [7] A. R. Its, A. G. Izergn, V. E. Korepn and N. A. Slavnov, Phys. Rev. Lett., [8] M. Shrosh, M. Takahash and Y. Nshyama, Emptness Formaton Probablty for the One-Dmensonal Isotropc XY Model, cond-mat/ [9] N. Ktanne, J. M. Mallet, N. A. Slavnov, V. Terras, Correlaton functons of the XXZ- Hesenberg chan at the free fermon pont from ther multple ntegral representatons, hep-th/00369, to appear n Nuclear Physcs B, 00. [0] N. Ktanne, J. M. Mallet, N. A. Slavnov, V. Terras, J. Phys. A: Math. Gen L385-L388; hep-th/0034. [] A. V. Razumov and Yu. G. Stroganov, J. Phys. A: Math. Gen ; condmat/004. 7

8 [] H. E. Boos, V. E. Korepn, Y. Nshyama, M. Shrosh, Quantum Correlatons and Number Theory, cond-mat/0034. [3] E. Leb, T. Shultz and D. Matts, Ann. Phys.,