Spectral properties of supersymmetric Polychronakos spin chain associated with A_{N-1} root system

Sp ectral p rop erties of sup ersy mmetric P olychronakos spin chain asso ciated w ith A N − 1 ro ot system B. Basu-Mallick ∗ and Nilanjan Bondyopadha y a † The ory Gr oup, Saha Institute of Nucle ar Physics, 1/AF Bidhan Nagar, K olkata 700 064, India Abstract By using the exact partition function of su ( m | n ) P olyc hronak os spin c hain asso ciated with A N − 1 ro ot sys tem, w e study some stat istical pr o p erties of the related sp ectrum. It is found t ha t the corresp onding energy lev el densit y satisfies the Ga ussian distribution and the cum ulativ e distribution of spacing b et w een consecutiv e energy lev els ob eys a certain ‘square ro ot of a loga r it hm’ law. P A CS : 02.30 .Ik; 75.10.Jm; 05 .30.-d; 75.10 .Pq Keywor ds : Exactly solv able quan tum spin c hains; Sup ersymme try; Partition function; Lev el densit y distribution ∗ e-mail address: bireswar.basumallick@saha.ac.in † e-mail address: nilanjan.b ondyopadhay a @saha.ac.in 1 Exactly solv able one dimensional quan tum integrable spin c hains and dynamical mo d- els with long-range interaction [1- 11] ha v e attracted muc h atten tio n in r ecent y ears due to their app earance in a wide rang e of sub jects like fractional statistics [12, 13], qu an- tum electric transp ort in mesoscopic systems [14], Y angian quan tum group [15- 17], SUSY Y ang-Mills theory a nd s tring theory [18 -20]. Among quan tum spin systems with long - range in teraction, the w ell kno wn spin- 1 2 Haldane-Shastry (HS) model is in tro duced in an attempt to construct a n exact ground state whic h w ould coincide with the U → ∞ limit of Gut willer’s v ar ia tional w av e function fo r the Hubbard mo del [4, 5]. A natural su ( m ) generalization of this exactly solv able HS mo del is constructed by using the ‘spin’ exc hange op erato r a sso ciated with the fundamen tal represen tation of su ( m ) algebra [6, 7]. Subsequen tly , it is realized that suc h HS spin chain may b e repro duced from the trigono- metric spin Calogero-Sutherland mo del b y applying the ‘freezing trick’, whic h basically uses the fact tha t spin and dynamical degrees of freedom of the latter mo del decouple from eac h other for large v alues of the coupling constant. F urthermore, a new quan tum spin chain with long-rang e in teraction is o btained b y applying this freezing tric k to the case of rational spin Calog ero mo del [8]. Lattice sites of this spin c ha in, whic h a r e in- homogeneously distributed o n a line, are determined through the zeros of the Hermite p olynomial [10]. This quantum in tegrable as w ell as exactly solv able spin system is usu- ally kno wn as P olyc hronak os o r P olyc hronak os-F rahm spin c hain in the literature. Both HS and P olychronak os spin c hains admit su ( m | n ) sup ersymmetric extensions, in whic h eac h site is o ccupied by either o ne of the m t yp e o f b osonic states or one of the n type of fermionic states [21-27]. Suc h sup ersymmetric spin chains pla y a role in describing some strongly correlated syste ms in condensed matter phys ics, where holes mo ving in the dynamical bac k ground of spins b eha v e as b osons, and spin-1/ 2 electrons b eha v e as fermions [28, 29]. It should be noted that, the ab o v e men tioned HS and P o lyc hronak os spin c hains (along with their sup ersymme tric extensions) are all related t o the A N − 1 t yp e of ro ot system, for 2 whic h the in teraction b et w een any t w o spins dep ends only on t he difference of their site co ordinates. V arian ts of these spin c ha ins associated with other ro ot systems , endow ed with more general type of interactions, ha v e also b een studied in t he literature [30-36]. By a pplying the metho d of freezing tric k, it has b ecome p ossible to compute the exact partition functions of su ( m ) P olychronak os and su ( m ) HS spin chains associated with A N − 1 ro ot syste m [9, 37], s u ( m | n ) sup ersymmetric extensions of these spin chains [23, 2 5], and v ariants o f such spin chains related to other ro ot systems [33-36]. These exact partition functions ha v e turned out to b e a very efficien t to o l for studying some statistical prop erties of the relat ed energy spectra lik e lev el densit y distribution a nd dis- tribution of spacing b etw een consecutiv e ene rgy lev els. It is found that, for sufficien tly large num b er of la t t ice sites, the energy lev el densit y of t his type of spin c hain follo ws the Gaussian distribution with high degree o f accuracy [33-38,25]. It is also observ ed that, distribution of spacing b et we en consecutiv e energy lev els for suc h in tegrable spin chains is not of P oisson t yp e, as may b e exp ected due to a w ell-kno wn conjecture of Berry a nd T abo r [39]. Ho w ev er it app ears that, ev en though the exact partition function of su ( m | n ) sup ersymm etric P o lyc hronak os spin chain associated with A N − 1 ro ot sys tem ha s b een deriv ed b y using the freezing tric k [2 3], statistical prop erties of t he related sp ectrum lik e distributions of energy leve l densit y and spacing b etw een consecutiv e energy lev els hav e not b een analyzed till now. The purp ose of this letter is to study these sp ectral prop erties of the sup ersymmetric Polyc hr o nak os spin chain by using its exact partitio n function. The Hamiltonian of the su ( m | n ) sup ersymmetric P olyc hro nak os spin c hain asso ciated with the A N − 1 ro ot system is given b y [2 3] H ( m | n ) = X 1 ≤ j > σ , (iv) E min and E max are approx imately symmetric w ith resp ect to the mean energy µ , namely | E max + E min − 2 µ | << E max − E min , then P ( s ) is approxim ately giv en b y an analytic expression of t he form [34] ¯ P ( s ) = 1 − 2 √ π s max r lo g  s max s  , (16) where s max denotes the maxim um normalized spacing, whic h can b e estimated with g reat accuracy through t he relation s max = E max − E min √ 2 π σ . (17) Studying t he case of su ( m | n ) sup ersymmetric P olyc hronak os spin c hain for a wide range of v a lues o f m , n and N , w e also find that the spacing b etw een consec utiv e energy lev els do es not follo w the Poiss on distribution. So it is natural to explore the applicabilit y of expression (16) for the presen t case. T o t his end, let us first c hec k whether the energy sp ectrum of su ( m | n ) sup ersymme tric P olyc hronak os spin c hain satisfies the four condi- tions that ha ve b een describ ed in the previous paragraph. W e ha v e already found that 10 this sp ectrum is equally spaced with unit in t erv al and the lev el densit y follo ws Ga ussian distribution with go o d approximation for sufficien tly large v alues of N . So this sp ectrum eviden tly satisfies the first t w o conditions. F rom Eq. (14) a nd the expressions of lo we st and highest energy levels giv en b y E min = 0 and E max = N ( N − 1) / 2 resp ectiv ely , it follo ws that b o t h ( µ − E min ) /σ and ( E max − µ ) /σ v ary a s √ N when N → ∞ . Therefore, this spectrum conforms to the t hird condition. At N → ∞ limit, it is also easy to find that | E min + E max − 2 µ | = X ( m, n )( E max − E min ) , (18) where X ( m, n ) = | m − n | ( m + n ) 2 . Note that X ( m, n ) b ecomes zero for m = n and ta k es a finite nonzero v alue f or m 6 = n . Conseque n tly , the fourth condition is satisfied only for the case m = n . Ho w eve r it can b e sho wn that, if one drops this for th condition, Eq. (16) still holds within a sligh tly smaller range of s [35]. Therefore, it is natural to exp ect t ha t P ( s ) w ould follow the a nalytical expression ¯ P ( s ) (16) in the case su ( m | n ) P olyc hrona kos spin c hain for all p ossible v alues o f m, n and sufficien tly large v alues of N . T o supp ort the ab ov e conclusion, w e study n umerically the na ture of P ( s ) with dif- feren t v alues of m , n and N . F or example, w e may consider the part icular case o f su (1 | 1) spin chain with N = 30 lattice sites. F or this case, P ( s ) and ¯ P ( s ) are drawn as the dott ed line and the contin uous line resp ectiv ely in F ig. 2 . F rom this figure it is eviden t that, the cum ulativ e distribution of spacing matc hes with ¯ P ( s ) extreme ly w ell. The MSE for this case is obtained as 2 . 628 × 10 − 6 . Next, we consider the case o f su (2 | 1) spin chain with N = 30 lattice sites. The corresp onding P ( s ) a nd ¯ P ( s ) a re plot t ed a s the dotted line a nd the contin uous line resp ectiv ely in Fig. 3 . Again, a v ery go o d agreemen t is found b et wee n these tw o lines with MSE give n by 4 . 724 × 10 − 4 . By studying suc h particular cases w e conclude tha t, similar to the case of ot her quan tum in tegra ble spin c hains with long-range interaction that ha v e b een studied so far, the cum ulativ e distribution of spac- ing b etw een consecutiv e energy leve ls of the supersymmetric P olychronak os spin ch ain follo ws the expression (1 6) with remark able a ccuracy . 11 References [1] F . Calogero, J. Math. Phy s. 12 (1971) 419. [2] B. Sutherland, Ph ys. Rev. A 4 ( 1971) 2 019; B. Sutherland, Ph ys. Rev. A 5 (19 72) 1372. [3] M.A. Olshanetsky and A.M. P erelomo v, Phys . Rep. 94 (1983 ) 313. [4] F .D.M. Haldane, Phys . Rev. Lett. 60 (19 8 8) 635. [5] B. S. Shastry , Ph ys. R ev. Lett. 6 0 (198 8) 639 . [6] N. Ka w ak a mi, Ph ys. R ev. B 46 ( 1 992) 1 005. [7] Z .N.C. Ha and F.D.M. Haldane, Ph ys. Rev. B 46 ( 1 992) 9359 . [8] A.P . Polyc hro nak os, Ph ys. Rev. Lett. 70 (1993) 23 2 9. [9] A.P . Polyc hro nak os, Nucl. Ph ys. B 419 (19 9 4) 553 . [10] H. F rahm, J. Ph ys. A 26 (1 993) L473. [11] Z.N.C. Ha, Quantum many-b o dy systems in on e dim ension, Series on Adv ances in Statistical Mec hanics, V ol.12 (W orld Scien tific,19 96). [12] M .V.N. Murth y and R . Shank ar, Phys . Rev. Lett. 73 (199 4) 3331. [13] A.P . P olyc hrona k os, Gener alize d statistics in one dimension , Les Houc hes 1998 lec- tures, hep-th/9902157. [14] C .W.J. Beenakk er and B. Reja ei, Ph ys. Rev. B 49 (19 94) 7499; M. Caselle, Ph ys. Rev. Lett. 74 (199 5 ) 2776. [15] F. D. M. Haldane, Z.N.C. Ha, J.C. T alstra, D . Benard and V. P asquier, Ph ys. Rev. Lett. 69 (1992 ) 2021. 12 [16] D. Benard, M. Gaudin, F. D. M. Haldane, and V. P a squier, J. Ph ys. A 26 (199 3 ) 5219. [17] K.Hik ami, Nucl. Ph ys. B 441 (19 95) 53 0. [18] D. Berenstein, S.A. Cherkis , Nucl. Ph ys. B 702 ( 2004) 49. [19] D. Serban, M. Staudache r, JHEP 04 0 6 (200 4) 001 . [20] N. Beisert, M. Staudac her, Nucl. Ph ys. B 727 (2005) 1. [21] F. D. M Haldane, in Pro c. 1 6th T aniguc hi Symp., Kashik o jima, Japa n (1993), eds. A. Okiji and N. Kaw ak ami (Springer, 1 994). [22] B. Basu-Mallic k, Nucl. Ph ys. B 540 (1 999) 679. [23] B. Basu-Mallic k, H. Ujino and M. W adati, Jour. Phys . So c. Jpn. 68 (19 99) 32 19. [24] K. Hik a mi and B. Basu-Mallic k, Nucl. Phys . B 566 (2000) 51 1 . [25] B. Basu-Mallic k and N. Bondyopadha y a, Nucl. Ph ys. B 757 (2006 ) 280. [26] B. Basu-Mallic k, N. Bondy opadha y a and D. Sen, Nucl. Phy s. B 79 5 (200 8) 596 . [27] R. Thomale, D. Sch uric ht and M. G reiter, Ph ys. Rev. B 74 (20 06) 02442 3 . [28] P . Sc hlottmann, In t. Jour. Mo d. Phys . B 11 (1997) 355 . [29] M . Arik a w a, Y. Saiga and Y. Kuramoto, Ph ys. Rev. Lett. 86 (2001) 3096; M. Arik aw a and Y. Saiga, J. Ph ys. A 39 (2 006) 1 0603. [30] D. Bernard, V. Pasq uier and D . Serban, Europhys . Lett. 30 (199 5) 301. [31] T . Y amamoto and O. Tsuc hiy a , J. Ph ys. A 29 (1 9 96) 39 77. [32] E . Corriga n and R. Sasaki, J. Ph ys. A 35 (2002) 7017. 13 [33] A. Enciso, F . Fink el, A. Gonz´ alez-L´ op ez a nd M.A. Ro dr ´ ıguez, Nucl. Ph ys. B 707 (2005) 553. [34] J. C. Barba, F. Finke l, A. Gonz´ alez-L´ op ez and M.A. Ro dr ´ ıguez, Phys . Rev. B 77 (2008) 214422. [35] J. C. Barba, F. Finke l, A. G o nz´ alez-L´ op ez and M.A. Ro dr ´ ıguez, Nucl. Phys . B 806 (2009) 684. [36] B. Basu-Mallic k, F. Fink el and A. Go nz´ alez-L´ op ez, arXiv: 0809.423 4, to b e published in Nuclear Ph ysics B. [37] F. Fink el and A. G o nz´ alez-L´ op ez, Ph ys. Rev. B 72 (200 5) 174 411. [38] J. C. Barba, F. Fink el, A. Gonz´ alez-L´ op ez and M.A. Ro dr ´ ıguez, Europh ys Lett. 83 (2008) 27005. [39] M . V. Berry a nd M. T ab or, Pro c. R. So c. Lo nd. A 356 (1 977), 375 [40] S. Ahmed, Lett. Nuov o Cimento 22 (1978) 367. [41] F. Haak e, Quantum signatur es of Chaos (Springer-v erlag, 200 1 ). 14 0 100 200 300 400 E i 0 0.004 0.008 0.012 D (1|1) (E i ), G(E i ) D (1|1) (E i ) G(E i ) Figure 1: Plot of the degeneracies D (1 | 1) ( E i ) v ersus energies E i (dotted line) for the case of su (1 | 1) Polyc hronak os spin c hain with N = 30, and its comparison with the Ga ussian distribution (contin uous line). 15 0 1 2 3 4 s 0 0.2 0.4 0.6 0.8 1 P(s), P(s) P(s) P(s) Figure 2: The dotted curv e represen ts the cum ula t ive distribution o f spacing b etw een consecutiv e energy lev els for su ( 1 | 1) spin c hain with N = 30, and the contin uous curv e represen ts the corresp onding ¯ P ( s ). 16 0 1 2 3 4 5 s 0 0.2 0.4 0.6 0.8 1 P(s), P(s) P(s) P(s) Figure 3: The dotted curv e represen ts the cum ula t ive distribution o f spacing b etw een consecutiv e energy lev els for su ( 2 | 1) spin c hain with N = 30, and the contin uous curv e represen ts the corresp onding ¯ P ( s ). 17