The Transmission Property of the Discrete Heisenberg Ferromagnetic Spin Chain

The T ransmiss ion Prop erty of the Discrete Heisen b erg F erromagnetic Spin C hain Qing Ding ∗ and W ei Lin Inst. of Math. and Key Lab. of Math. f or Nonlinear Scie nces F udan Univ ersit y , Shanghai 200433, P .R. China Abstract W e present a mechanism f or displa ying t he transmission prop erty of the disc r ete Heisenberg ferromag netic spin ch ain (DHF) via a geometric appro ach. By the aid of a discrete nonlinear Sc hr¨ odinger-like equation which is the discrete gauge equiv alent to the DHF, we show that the det ermina tion of trans mitting co efficie nts in the tra nsmis- sion problem is alwa ys bistable. Thus a definite alg o rithm and gener a l stochastic algo - rithms are pr esented. A new in v aria nt p erio dic phenomenon of the non-trans mitting behavior for the DHF, with a large probability , is revealed by a n ado ption o f v arious sto chastic a lg orithms. P ACS num b ers: 02.40 .Ky; 05 .45.Mt; 07 .55.Db § 1 In tro duction There is curren t in terest in displa ying the prop erties of one-dimensional magnetic mo d - els. The one-dimensional classical con tin uu m Heisenb er g mod els with different magnetic in teractions ha ve b een settled as one of the inte resting and attractiv e cl ass of nonlin- ear dynamical equations exhibiting the complete int egrabilit y on man y o ccasions ([1]-[7]). Ho w ev er, th ou gh the in vesti gation of nonlinear spin c hain systems is quite fascinating, few has b een known in the case of m ore realistic ph ysical lattice s p in c hains so far, es- p ecially , for the follo w ing d iscr ete (isotropic) Heisen b erg f er r omagnetic spin c hain with nearest n eigh b or exc hange interac tion (DHF), ˙ S n = S n × ( S n +1 + S n − 1 ) , (1) where S n = ( s 1 n , s 2 n , s 3 n ) ∈ R 3 with ( s 1 n ) 2 + ( s 2 n ) 2 + ( s 3 n ) 2 = 1 and th e dot stands for the time deriv ativ e. In fact Eq.(1) comes from the Hamiltonian formalism: ˙ S n = { S n , H } with the Hamiltonian function H = P + ∞ j = −∞ S j · S j +1 , where · d enotes the inn er pro duct of v ectors in R 3 , and th e Poi sson brac ke t { s a j , s b k } = δ j k P 3 c =1 ε abc s c j , where δ j k is Kronec ker’s ∗ qding@fudan.edu.cn 1 sym b ol and ε abc is the 3-dimen s ional totally an tisymmetric s ym b ol. Physically , one w ould sa y the mo del (1) describ es a system of classical spin s and su b jects to homogenous nearest- neigh b or Heisen b erg inte raction. The standard con tinuous limit pro cedur e p erformed on Eq.(1) leads to the inte grable Heisen b erg ferromagnetic mo del: S t = S × S xx , which is an imp ortant equ ation on condensed matter ph ysics (see, for example, [8]). Eq.(1) is quite w ell known and there is little need to giv e here a detailed enumeration. It should b e also emphasized that Eq.(1) is w idely b eliev ed to b e not a completely integ rable equation. T o our b est kno wledge, there is no effectiv e metho d in study of the dynamical b eha viors of Eq.(1) in literature. On the other hand, the geometric concept of gauge equiv alence [9, 10] b et w een in te- grable equations, whic h pro vides a useful to ol in displaying solitonic d ynamics, has b een generalized to noninte grable case in [11, 12]. In [12] it is shown that the discrete nonlinear (nonin tegrable) Schr¨ odinger equation (AL-DNLS) is the discrete gauge equiv alen t to a nonint egrable discrete Heisen b erg mo del and the transmission prop erties of the AL-DNLS equation ([13 ]-[18]) are completely preserved under the action of discrete gauge transfor- mations. In this pap er, w e wo uld app ly this geometric idea to transform the DHF (1) in to a discrete nonlinear Sc hr¨ odinger-like equation and utilize the nonlinear S c hr¨ odin ger-lik e equation to study the transmission p roblem of the DHF (1) fr om m athematical p oin t of view. W e hop e that this stud y will b e helpful to rev eal deep er qu an tum chaoti c pr op erties of th e DHF (1) and useful in physical applications. This article is organize d as follo ws. In sectio n 2 w e deduce a discrete n onlinear Sc h¨ odinger-type equation whic h is the discrete gauge equiv alent to the DHF in the cate- gory of the (discrete) Y ang-Mills theory . In section 3 we displa y the transmission p r op ert y for the DHF (1) with th e aid of the discrete nonlinear S c h¨ odinger-t yp e equation and an appro ximate linear s tability analysis for the stationary v ersion of the discrete nonlinear Sc h¨ odinger-type equation is giv en for s upp orting the discussion of the transm ission exp o- sitions. Finally , in section 4, we close the pap er with some conclusions and remarks. § 2 Gauge equiv alence F ollo w ing th e concept of discrete conn ection and asso ciated discr ete curv ature introd uced in [12], we wo uld express Eq.(1) geometrica lly as a discrete n onlinear equation with pre- scrib ed d iscrete curv ature representa tion. W e fir st con ve rt Eq.(1) into its matrix v ersion: ˙ S n = − i [ S n , S n +1 ] 2 − i [ S n , S n − 1 ] 2 , (2) where S n = s 1 n s 2 n − is 3 n s 2 n + is 3 n − s 1 n ! . Th en we define a discrete connection { A n } by A n = ( L n , M n ) , (3) 2 where L n = z + z − 1 2 I + z − z − 1 2 S n , M n = i  1 − z 2 + z − 2 2  S n + S n − 1 2 − i z 2 − z − 2 2 I + S n − 1 S n 2 with I denoting the 2 × 2 iden tit y matrix and z b eing a free sp ectral p arameter. It is a direct and length y computation, in the s im ilar wa y displa y ed in the app end ix of [12], that the corresp ondin g d iscr ete curv ature { F A n } is giv en b y F A n := ˙ L n − M n +1 L n + L n M n = z − z − 1 2  ˙ S n + i [ S n ,S n − 1 ] 2 + i [ S n ,S n +1 ] 2  + i − z 3 + z + z − 1 − z − 3 4 ( S n · S n − 1 − S n · S n +1 ) S n + i − z 3 + z − z − 1 + z − 3 4 ( S n · S n − 1 − S n · S n +1 ) I . Here we hav e u sed the identi ty: S n +1 S n = − S n S n +1 + ( S n · S n +1 )2 I and similarly S n S n − 1 = − S n − 1 S n + ( S n · S n − 1 )2 I in the com- putation. Th us w e see that, if we set K n = i − z 3 + z + z − 1 − z − 3 4 ( S n · S n − 1 − S n · S n +1 ) S n + i − z 3 + z − z − 1 + z − 3 4 ( S n · S n − 1 − S n · S n +1 ) I , Eq.(2) is equ iv alen t to holding the f ollo wing pre- scrib ed d iscrete curv ature representa tion: F A n := ˙ L n − M n +1 L n + L n M n = K n , ∀ n. (4) Our aim in th is section is to geometrically trans f orm Eq.(2) (resp ., the connection (3)) to a d iscrete nonlinear Sc hr¨ odin ger-lik e equation (resp., a new connection) by a discrete gauge sequence { G n } . So the k ey p oin t is to fi nd su c h a { G n } . F or this purp ose, f or a matrix sequence { S n } solving Eq.(2), w e shall fin d { G n } to satisfy σ 3 = − G n S n G − 1 n , G n +1 = 1 ¯ q n − q n 1 ! G n for some (complex v alued) q n , (5) where σ 3 = 1 0 0 − 1 ! is the P auli matrix. The fir s t equation of (5) h as a class of solutions of the form G n = i p 2(1 − s 1 n ) ( σ 3 − S n )diag( F n , ¯ F n ) , (6) where { F n } is free and to b e sp ecified latter. It is a straigh tforw ard v erification that, in order for (6) to fulfill th e second equation of (5) for some complex sequence { q n } , F n is forced to satisfy the follo wing iterated relation (1 − s 1 n )(1 − s 1 n +1 ) F n +1 F − 1 n +( s 2 n − is 3 n )( s 2 n +1 + is 3 n +1 ) F n +1 F − 1 n = 2 q (1 − s 1 n )(1 − s 1 n +1 ) , ∀ n. So we may determine F n progressiv ely by this relation in n and hence ha v e pro v ed th e existence of a desired gauge sequence { G n } . F or the connection (3), we use { G n } to define a new connection { A G n = ( L G n ( t, z ) , M G n ( t, z )) } b y L G n ( t, z ) = G n +1 L n G − 1 n = G n +1  z + z − 1 2 I + z − z − 1 2 S n  G − 1 n = z − 1 z ¯ q n − z − 1 q n z ! and M G n ( t, z ) = ˙ G n G − 1 n + G n M n G − 1 n = ˙ G n G − 1 n + i ( − 1 + z − 2 ) 1 1+ | q n − 1 | 2 (1 − z 2 ) ¯ q n − 1 1+ | q n − 1 | 2 (1 − z − 2 ) q n − 1 1+ | q n − 1 | 2 (1 − z 2 ) 1 1+ | q n − 1 | 2 ! . Here w e ha v e used the relations (5) in the computation. Since L n , M n satisfy the p rescrib ed discrete curv ature representati on (4), from Lemma 1 pro ve d in [12] we kno w that L G n , M G n should fu lfill ˙ L G n − M G n +1 L G n + L G n M G n = G n +1 K n G − 1 n , ∀ n. (7) 3 It is a direct compu tation that l.h.s. of (7) = 1 ¯ q n − q n 1 ! " z − 1 0 0 z ! ˙ G n G − 1 n − ˙ G n G − 1 n z − 1 0 0 z !# + i   a n − ( z − 1 − z 3 ) ¯ q n 1+ | q n | 2 + ( z − z 3 ) ¯ q n +( z − 1 − z ) ¯ q n − 1 1+ | q n − 1 | 2 − ( z − z − 3 ) q n 1+ | q n | 2 + ( z − 1 − z − 3 ) q n +( z − z − 1 ) q n − 1 1+ | q n − 1 | 2 b n   (8) where a n = ( z − 1 − z − 3 ) − ( z − z − 1 ) | q n | 2 1+ | q n | 2 + ( − z − 1 + z − 3 )+( z − z − 1 ) ¯ q n q n − 1 1+ | q n − 1 | 2 and b n = − ( z − z 3 )+( z − z − 1 ) | q n | 2 1+ | q n | 2 + ( z − z 3 )+( z − z − 1 ) q n ¯ q n − 1 1+ | q n − 1 | 2 and r.h.s. of (7) = i  1 1 + | q n | 2 − 1 1 + | q n − 1 | 2  z − 1 − z − 3 ( − z + z 3 ) ¯ q n ( − z − 1 + z − 3 ) q n − z + z 3 ! . (9) Here we hav e used the relation (5) and some ident ities display ed in the app endix of [12]), e.g., S n · S n +1 = 1 −| q n | 2 1+ | q n | 2 and S n · S n − 1 = 1 −| q n − 1 | 2 1+ | q n − 1 | 2 , in the ab o ve computations. Thus, by substituting (8) and (9) into (7), the equation of the off-diagonal p art of (7) leads to ˙ G n G − 1 n = ∗ i ¯ q n 1+ | q n | 2 − i ¯ q n − 1 1+ | q n − 1 | 2 i q n 1+ | q n | 2 − i q n − 1 1+ | q n − 1 | 2 ∗ ! , (10) where ∗ are some expressions w hic h cannot b e carried out at this moment. On th e other hand, at the same time w e also h a v e l.h.s. of (7) = 0 ˙ ¯ q n z − ˙ q n z − 1 0 ! + z − 1 ¯ q n z − q n z − 1 z ! ˙ G n G − 1 n − ˙ G n +1 G − 1 n +1 z − 1 ¯ q n z − q n z − 1 z ! + i   a n − ( z − 1 − z 3 ) ¯ q n 1+ | q n | 2 + ( z − z 3 ) ¯ q n +( z − 1 − z ) ¯ q n − 1 1+ | q n − 1 | 2 − ( z − z − 3 ) q n 1+ | q n | 2 + ( z − 1 − z − 3 ) q n +( z − z − 1 ) q n − 1 1+ | q n − 1 | 2 b n   . (11) By sub stituting (11) and (9) into the equation (7) and com binin g w ith (10), th e diagonal part in this time implies ˙ G n G − 1 n = α n i ¯ q n 1+ | q n | 2 − i ¯ q n − 1 1+ | q n − 1 | 2 i q n 1+ | q n | 2 − i q n − 1 1+ | q n − 1 | 2 σ n ! (12) where α n and σ n satisfy    α n − α n +1 = i  ¯ q n q n − 1 1+ | q n − 1 | 2 − ¯ q n +1 q n 1+ | q n +1 | 2  σ n − σ n +1 = i  q n +1 ¯ q n 1+ | q n +1 | 2 − q n ¯ q n − 1 1+ | q n − 1 | 2  . (13) Ob viously , the second equation of (13) has a solution σ n = ¯ α n . By su bstituting (12) and (9) into (7), the off-diagonal part leads to { q n } satisfying the follo wing discrete nonlinear Sc hr¨ od in ger-lik e equation ˙ q n + i q n +1 1 + | q n +1 | 2 − 2 i q n 1 + | q n | 2 + i q n − 1 1 + | q n − 1 | 2 + ( α n − ¯ α n +1 ) q n = 0 , (14) 4 where { α n } solv es the fir st equation of (13). This sh o ws that the DHF (1) is the (discrete) gauge equ iv alen t to the discrete n onlinear Schr¨ odinger-lik e equation (14). W e wo uld p oin t out that Eq.(14) sets { A G n = ( L G n ( t, z ) , M G n ( t, z )) } as its discrete connection and has p rescrib ed discrete cur v ature r epresen tation: F G n = K G n , where K G n is giv en by righ t-hand-sid e of (9). Notice that the expression K G n is to b e zero at z = 1, it can also b e pr o v ed con v ersely that the d iscrete n onlinear Sc hr¨ odinger-lik e equation (14) is (discrete) gauge equiv alen t to the DHF (1) by the gauge sequence { G n } satisfying G n +1 = L G n ( t, 1) G n , ˙ G n = M G ( t, 1) G n , ∀ n (the existence of su c h a sequence { G n } is b ecause of the integrabilit y of { A G n } , or in other words its zero curv ature repr esen tation, at z = 1). The d etails are omitted here and one may refer to [12] for a reference. { α n } app eared in Eq.(14) can b e solv ed out from the fi rst equation of (13) as follo ws in d ifferen t tw o app roac hes. On e is obtained b y iterating fr om initial data q 0 , q − 1 and α 0 : α n =              i ¯ q n q n − 1 1+ | q n | 2 + i P n k =1  1 1+ | q k − 1 | 2 − 1 1+ | q k − 2 | 2  ¯ q k − 1 q k − 2 − i ¯ q 0 q − 1 1+ | q − 1 | 2 + α 0 , n > 0 α 0 , n = 0 i ¯ q n q n − 1 1+ | q n | 2 + i P − 1 k = n  1 1+ | q k | 2 − 1 1+ | q k +1 | 2  ¯ q k +1 q k − i ¯ q 0 q − 1 1+ | q 0 | 2 + α 0 , n < 0 . (15) The other is obtained by iterating f rom the b oun dary data at p ositiv e infinity: α n = i + ∞ X k = n  1 1 + | q k | 2 − 1 1 + | q k +1 | 2  ¯ q k +1 q k + i ¯ q n q n − 1 1 + | q n − 1 | 2 ∀ n. (16) This refl ects that th e pr op ert y of th e nearest n eighb or exchange in teraction for Eq.(1) col- lapses for its gauged equiv alen t equation (14), since th e n -th c hain’s exc hange interactio n in Eq.(14) relates to chains from − 1-st chain to ( n ± 1)-th chain or all the c hains with lab el > n − 1. Though th is implies that the dyn amics of Eq .(14 ) and hence the original Eq.(1) will b e complicated, the exp osition of the appr o ximate linear stabilit y in the next section for the stationary version of Eq.(14) shows in interesting fact: the n -th c hain term is related to its nearest neigh b or exchange in teraction in the stabilit y analysis. This reflects that Eq.(14) do es h a v e the p rop erty of the nearest neighb or exc hange interactio n in some sense. W e note that ([4, 7, 12]) the discrete nonlinear Sc hro d inger equ ations without non constant denominator terms (e.g. th e AL equ ation: i ˙ q n + ( q n +1 + q n − 1 − 2 q n ) + | q n | 2 ( q n +1 + q n − 1 ) = 0) are gauge equiv alent to the (mo d ified) discrete Heisen b erg spin c hain mo dels with n on- constan t denominator terms (e.g., the Ishimori equation: ˙ S n = 2 S n +1 × S n 1+ S n +1 · S n + 2 S n × S n − 1 ] 1+ S n · S n − 1 ). And, meanwhile, the ab o v e exp osition indicates th at the discrete Heisen b erg spin c hain mo del (1 ) without nonconstant denominator terms is gauge equ iv alen t to the discrete nonlinear Sc hro d inger-t yp e equation (14) with nonconstant denominator terms. Th is is an interesting du alit y phenomenon b et wee n discrete nonlin ear Schrod inger-t yp e equations and d iscrete Heisenberg spin c hain mo d els. Before w e use (14) to study the transmission pr oblem of the DHF, let’s giv e a general description of constructing solutions to th e DHF (1), or equ iv alen tly (2), from those to 5 the d iscrete nonlin ear S c hr¨ odinger-lik e equation (14). F or a solution { q n } to Eq.(14), let G 0 b e a fu ndamenta l s olution to ˙ G 0 = M 0 ( t, 1) G 0 = i α 0 ¯ q 0 1+ | q 0 | 2 − ¯ q − 1 1+ | q − 1 | 2 q 0 1+ | q 0 | 2 − q − 1 1+ | q − 1 | 2 ¯ α 0 ! G 0 . (17) It is a direct verificatio n that, by successiv e iteration, G n = ( L G n − 1 ( t, 1) G n − 1 , n > 0 L G n +1 ( t, 1)) − 1 G n +1 , n < 0 = ( L G n − 1 ( t, 1) · · · L G 0 ( t, 1) G 0 , n > 0 ( L G n +1 ( t, 1) · · · L G 0 ( t, 1)) − 1 G 0 , n < 0 (18) solv es G n +1 = L G n ( t, 1) G n , ˙ G n = M G ( t, 1) G n , ∀ n . Therefore { S n = − G − 1 n σ 3 G n } is a solu- tion to the DHF (2) w hic h corresp onds to { q n } u nder the discrete gauge transformation. § 3 App lications In this section, we shall use the gauged equ iv alen t equation (14) to stud y dynamical prop erties of the discrete Heisen b erg f erromagnetic spin c hain (1). W e m ainly focu s on exploring the transmission p rop ert y of the DHF (1 ) and its r elated linear stabilit y analysis for the stationary ve rsion of the equation (14). A T ransmission prop ert ies In this sub section, we stud y as a physical app licatio n w hether the w a ve transmiss ion prop erty ([13, 16]) of the discrete nonlin ear Sc hr ¨ odinger-lik e equation (14) and transfer it to that of the DHF (1) under the action of discrete gauge transformations. In order to do these, let’s consider the f ollo win g recurrence equation originated from the discrete nonlinear S c hr¨ odinger-like equation (14) b y setting q n ( t ) = ϕ n exp( − iE t ): − E ϕ n + ϕ n +1 1 + | ϕ n +1 | 2 − 2 ϕ n 1 + | ϕ n | 2 + ϕ n − 1 1 + | ϕ n − 1 | 2 − i ( α n − ¯ α n +1 ) ϕ n = 0 , (19) where ϕ n is a complex amplitude indep endent of th e time v ariable t , E is a real parameter and α n indep end en t of the time v ariable t solv es the fi rst equatio n of (13), wh ic h reads from (16), α n = i + ∞ X k = n  1 1 + | ϕ k | 2 − 1 1 + | ϕ k +1 | 2  ¯ ϕ k +1 ϕ k + i ¯ ϕ n ϕ n − 1 1 + | ϕ n − 1 | 2 ∀ n. (20) W e thus study the transmission problem of Eq.(19 ) via a similar numerical wa y of the AL-DNLS lattice c hain d ispla y ed in [13, 16] (please r efer to their pap ers for details). First w e note that E q .(19) has a stationary s olution as follo ws: ϕ n = T e iκn , n ∈ Z (21) 6 where th e real parameter E satisfies the consisten t cond ition 2 cos κ = 2 + E (1 + T 2 ) 1 + T 2 . (22) W e n o w consider the p r oblem: A finite nonlinear segmen t 0 6 n 6 N − 1 of length N in the nonnegativ e stationary regime is em b edd ed in a nonlinear chain { ϕ n } n ∈ Z satisfying (19) w ith ϕ n = ( R 0 e iκ + R e − iκ , n = 0 T e iκn , n > N , (23) where (22) is fu lfilled. Th is can b e r egarded as that an incident p lane wa ve R 0 e iκ on the left ( n = 0) indu ces a reflected plane w a v e R e − iκ on the left and a transmitted plane w a v e T e iκn on the righ t ( n > N ). R 0 is called the amplitude of the incoming wa ve, R the amplitude of the reflected wa ves and T the tran s mitted amplitude at th e right end of the nonlinear c hain; κ is called the out-coming w a ve num b er; | R 0 | 2 and | T | 2 are also called the in-coming and the transmitted intensit y resp ective ly . T he medium is completely nonlinear, th us the transm ission co efficien t T as a function of R 0 ma y n ot uniquely determined. If this is tru e, according to the sense m ade in [17, 13], it is called b istabilit y . F or the forwa rd transmission pr oblem of th e AL-DNLS lattice c hain, there o ccurs exactly the bistabilit y phenomenon ([13, 16]), wh ich leads Dely on et al [13] to consider th e bac kw ard transm itted problem for the DNLS. They p ro v ed that the pair ( κ, | T | ) initializes the inciden t in tensit y | R 0 | completely and then d isp la y ed th e transmission b eha vior of the DNLS in the ( κ, | T | ) plane. No w we consider the similar bac kward transmission pr oblem for solutions with the t yp e (23) to the present equ ation (19). F ollo wing W an and Souk oulis [15], we u se p olar co ordinates for ϕ n , that is ϕ n = R n e iθ n . F or a giv en pair ( κ, | T | ) (w ith ou t loss of the gen- eralit y , we may assume T > 0 in this pap er), w e try to iterate Eq.(19) from n = N to −∞ successiv ely and to determine the amplitudes ( R N − 1 , · · · , R 0 ) and p hases ( θ N − 1 , · · · , θ 0 ). The existence of solutions w ith the t yp e (23) to Eq.(19) w ill b e sh o wn b elo w in the algo- rithms. In ord er to f u rther su pp ort the existence of such solutions, an approximat e linear stabilit y analysis around th e the stationary solution (21) will b e give n in the next section, whic h fu rther reve als the linear stabilit y b eha viors of the equation (19). In this iterating pro cess, w e find that there are tw o choice s in determinin g the amplitudes ( R N − 1 , · · · , R 0 ) in eac h step. T h is implies that, not lik e the transmission pr ob lem of th e AL-DNLS, the pair ( κ, T ) d o es not initialize the incident intensit y | R 0 | completely , or in other words, th e bac kw ard transmission pr oblem for the equation (19) with solutions of type (23) is still bistable. W e first in tro du ce a definit e algorithm suc h that the pair ( κ , T ) initializes the am- plitudes ( R N − 1 , · · · , R 0 ) and p h ases ( θ N − 1 , · · · , θ 1 ) completely . In fact, such an algorithm is designed as follo ws . F rom ϕ n = T e iκn , n > N we get α n = i ¯ ϕ n ϕ n − 1 1+ | ϕ n − 1 | 2 , n > N from 7 (20) and the equation (19) for n > N + 1 is n o w equ iv alen t to (22). The equation (19) for n = N is rewritten as    T cos( κ ) + 1+ T 2 1+ R 2 N − 1 R N − 1 cos(∆ θ N ) = 2+ E (1+ T 2 ) 1+ T 2 T , T sin( κ ) − 1+ T 2 1+ R 2 N − 1 R N − 1 sin(∆ θ N ) = 0 , where ∆ θ n = θ n − θ n − 1 (w e r emark that θ N = κN ), w hic h leads R N − 1 to satisfying T = 1+ T 2 1+ R 2 N − 1 R N − 1 and θ N − 1 = κ ( N − 1). Since th e quadr atic equation T = 1+ T 2 1+ x 2 x in x j u st h as tw o self-recipro cal solutions: x = T an d x = 1 T , we then d etermine R N − 1 according to t he rule : to choose R N − 1 > 1 wh en 0 < T < 1 and R N − 1 < 1 wh en T > 1 in order to av oid getting the stationary solution (21). Next, from (20) we see that, f or n < N , α n = i P N k = n +1  1 1+ | ϕ k − 1 | 2 − 1 1+ | ϕ k | 2  ¯ ϕ k ϕ k − 1 + i ¯ ϕ n ϕ n − 1 1+ | ϕ n − 1 | 2 . Then the equation (19) for n < N is is equiv alen t to                  R n − 1 1+ R 2 n − 1 cos(∆ θ n ) = R n 1+ R 2 n ( E (1+ R 2 n )+2 1+ R 2 n − R n +1 ( 1 R n − R 3 n +2 R n +2 R n R 2 n +1 ) (1+ R 2 n )(1+ R 2 n +1 ) cos(∆ θ n +1 ) − 2 P N − 1 k = n +1  1 1+ R 2 k − 1 1+ R 2 k +1  R k R k +1 cos(∆ θ k +1 ) ) , R n − 1 1+ R 2 n − 1 sin(∆ θ n ) = R n +1 1+ R 2 n +1 cos(∆ θ n +1 ) . (24) The ab o v e exp osition indicates that R n − 1 satisfies R n − 1 1 + R 2 n − 1 = p (r.h.d of the fir s t eq.( 24 )) 2 + (r.h.d of the second eq.(24)) 2 and thus also has tw o self-recipro cal solutions. Ou r rule is that in the second step and the follo wing steps, we alwa ys c ho ose the ro ot with R n > 1 for the quadratic equations. According to this algorithm, we may uniquely determine the amp litud es and ph ases f rom n = N − 1 to −∞ successiv ely and esp ecially the incident inte nsity | R 0 | 2 . This pro cedu re also ind icates the existence of solutions of typ e (23) to th e stationary equ ation (19). If the resulting incoming wa ve intensit y | R 0 | 2 is of the same order of 1 | T | 2 (the recipro cal transmitted int ensity | T | 2 ) ind ep endent of N when 0 < T < 1, or is of the same order of | T | 2 when T > 1, w e say that the n on lin ear c hain with wa ve num b er k and outgoing in tensit y | T | 2 is to b e transmitting. If R 0 app ears to b e a rapid ly increasing function of N , we say that this non lin ear c hain is to b e non-transmitting. Figs.1(a) and (b) b elo w displa y the transmission b eha viors in the ( k , T ) p arameter p lane with the c hain length N = 110 and 150, resp ectiv ely , repr esen ting region of transm itting (white) and non- transmitting (blac k) b eha viors. In particular, Fig.1 (a) and (b) n umerically sho w the whole-scop e of the trans m itting b eha viors for κ ∈ (1 , 3 . 5) and T ∈ (0 . 5 , 2). The non- transmitting r egions form some strange patterns. The enlargement of one of these patterns, ( κ, T ) ∈ (1 . 3 , 1 . 7) × (1 , 1 . 4) , is further sho wn by Fig.1(c ). It is seen fr om th e figure that the t wisting b oun daries b et ween the transmitting r egion and the non-transmitting r egion 8 in the parameters plane alw a ys d ispla y sp iral, canto r-like , and fractal structure, whic h is v ery analogous to the structure of the standard chao tic attractor. F urthermore, from the Fig.1 (a) and (b), we see th at the transm itting b ehaviors only d ep end seriously on the parameters κ and T but d o not dep end sensitiv ely on the c hain length N . Figur e 1 ar ound her e. Ple ase find it at the end of the p ap er. Besides the defi n ite algorithm, we come to consider a sto c hastic algorithm . In such a sto c hastic algorithm, the determination of the bi-v alued R N − 1 should b e the same as that in the definite algorithm in order to a v oid getting a stationary solution; h ow ever, th e latter determin ations of R n (0 6 n 6 N − 2) in eac h step are based on the generation of a random num b er ξ , i.e., R i = R − if ξ > c ; R i = R + otherwise. Naturally , our fi rst c hoice of the generation of the random num b er ξ is the normal d istr ibution and the v alue c is tak en as zero, the exp ectatio n of this d istribution. It could b e seen from Fig.2 (a) that non- transmitting r egions (blac k) p erio dically app ear w ith resp ect to κ , here the c hain length is still to b e N = 150. Also, this kind of phenomena could b e somewhat found in Fig.1 (a) and (b). S econdly , a Poisson distrib ution with an exp ectation 5 is adopted to pro d uce the ran d om num b er ξ and c is tak en as 5. In this case, a similar p erio dic non-transm itting b eha vior is displa ye d in Fig2.( b). Actually , thousan d s of n umerical sim ulations v erify a fact that no matter wh at kin d of s to chastic distribution and c hain length are tak en, the non-transmitting b eh a vior is so prev alent around the v alues κ k ≈ 1 . 57 + k π 2 ( k = 0 , 1 , · · · ). This p erio d ic phenomena could b e regarded as an in v ariant non-transmitting b eha vior for Eq.(19). Figur e 2 ar ound her e. Ple ase find it at the end of the p ap er. The transmission problem asso ciated with the d iscrete Heisenberg ferromagnetic spin c hain (1) (or equiv alen tly (2)) is no w prop osed as follo ws. Th er e is a semi-infin ite nonlinear matrix wa ve chain { S n = − G − 1 n σ 3 G n } embedd ed in a nonlinear c hain of the DHF (2), where G n lo oks lik e (fr om (18)): G n = 1 ¯ q n − 1 − q n − 1 1 ! · · · 1 ¯ q 0 − q 0 1 ! G 0 , n > 1 (25) with G 0 satisfying (17) and q n = ϕ n e − iE t , where ϕ n is indep endent of t and giv en by (23). This nonlinear matrix wa v e chain is from the left tow ards the right , wh er e they are scattered in to reflected and transmitted parts. Similarly , R 0 , R present the amplitud es of the incoming and reflected w a v es and T the transmitted amplitude at th e right end of the nonlinear c hain und er consideration. | R 0 | 2 and | T | 2 are also called the in coming wa ve in tensit y and the transm itted in tensit y of the n onlinear m atrix c hain resp ectiv ely . As w e ha v e pro ved in the previous sectio n that the DHF (1) (i.e. the DHF (2)) is the d iscrete gauge equiv alent to the d iscrete nonlinear S c hr¨ odinger-like equ ation (14) and ve rse visa, w e see that the transmission prop erties of the equation (19) just d ispla y ed, i.e., the ab o v e definite algorithm and sto c hastic algorithms , are transferr ed completely to the trans- mission problem (25) of the DHF (2) und er the action of discrete gauge trans formations. 9 Th us th e nonlinear matrix wa ve c hain (25) of the DHF (2) h as the same trans mission b eha viors as those of nonlinear chain (23) of Eq.(19). Fig.1(a ),(b) and (c) are completely suitable in disp la ying the transmission b eha viors of the fin ite nonlinear matrix c hain of the DHF (2) in the parameters ( k , T ) plane with c hain length N = 110 and 150 un der the defi nite algorithm and so do Fig.2 (a) and (b) un der sto chasti c algorithms. F u rther- more, from the Fig.1 (a) and (b), w e see that the transmitting b ehaviors for solutions { S n = − G − 1 n σ 3 G n } with { G n } satisfying (25) to the DHF (1) only dep end s eriously on the parameters κ and T bu t do not d ep end sensitive ly on th e c hain length N . This sho ws that the imp ortant p henomenon of bistabilit y o ccurs still in sp in m agnetic physic s. The p erio dic phenomenon of the non-transmitting regions disp la y ed in Fig.2 provides a new and interesting transmission p rop ert y of the DHF (1). B Stabilit y analysis In order to sup p ort the exp osition in su bsection A and sho w the existence of stable so- lutions to (19) around the stationary solution (21), w e shall giv e an approximat e linear stabilit y analysis for Eq.(19). In w hat follo ws, we fo cus on the lo cally linear s tability of the stationary solution (21) in the rev erse iterativ e pro cedure through the linear v ariational metho d [19, 20]. F or this pu r p ose, we naturally imp ort the formulas as follo ws: R n = T + τ n , θ n = nκ + δ n , n = N , N − 1 , · · · , 2 , 1 , where qu an tities τ n and δ n , r esp ectiv ely , are regarded as small p ertur bations of the mo d- ulus and argumen t of the stationary solution (21). S ubstitute the ab o v e form ulas into Eq. (24) and then expand those nonlinear term s in the vicinit y of T and nκ . A tedious calculatio n thus giv es the follo wing equations at O ( τ i ) and O ( δ i ) ( i = n − 1 , n, n + 1):                                (1 − T 2 ) cos κ (1+ T 2 ) 2 τ n − 1 = 1 − T 2 (1+ T 2 ) 2 h E + 2 1+ T 2 − cos κ 1+ T 2 − T 2 cos κ 1+ T 2 i τ n + T 1+ T 2 h − 4 T (1+ T 2 ) 2 + cos κ T (1+ T 2 ) − 2 T (1 − T 2 ) cos κ (1+ T 2 ) 2 i τ n + T 1+ T 2 h − (1 − T 2 ) cos κ T (1+ T 2 ) 2 − − 2 T cos κ 1+ T 2 + 2 T (1 − T 2 ) cos κ (1+ T 2 ) 2 i τ n +1 + 2 T 4 cos κ (1+ T 2 ) 2 P N − 1 k = n +1 ( τ k − τ k +1 ) , sin κ 1+ T 2 ( δ n − δ n − 1 ) = − sin κ 1+ T 2 ( δ n +1 − δ n ) , (1 − T 2 ) si n κ (1+ T 2 ) 2 τ n − 1 = (1 − T 2 ) si n κ (1+ T 2 ) 2 τ n +1 , T cos κ 1+ T 2 ( δ n − δ n − 1 ) = T cos κ 1+ T 2 ( δ n +1 − δ n ) , (26) where n = N , N − 1 , · · · , 2 , 1. C learly , combining the second and the fou r th equations in (26) leads to δ n +1 − δ n = 0 , whic h shows an identit y of the series { δ n } at O ( δ n ). Hence, the linear stabilit y of the argumen t of the s tationary s olution in the reverse iteration pr o cedure b elongs to the stable case. Ho w eve r, the n onlinear stabilit y of this argument b ecomes a critical case, dep en ding 10 on the fu rther calculation at O ( δ 2 i ). Due to the tediousness of notations, w e omit the calculatio n here. Analogously , com binin g the first and the thir d equations in (26) yields: τ n − 1 =  1 + T 4 1 − T 4 − 2 T 2 (1 − T 4 ) cos κ  τ n − 2 T 4 1 − T 4 τ N . Hence, it follo ws that for n = N , · · · , 1, τ n − 1 = (  1 + T 4 1 − T 4 − 2 T 2 (1 − T 4 ) cos κ  N − n +1 − 2 T 4 1 − T 4 N − n X j =0  1 + T 4 1 − T 4 − 2 T 2 (1 − T 4 ) cos κ  j ) τ N , λ N ,n τ N . Consequent ly , the linear stabilit y of the mo d ulus of the stationary solution is stable in th e rev erse iteration provided that λ N ,n is u niformly b ounded for all N and n . More p recisely , the rev erse stabilit y holds wh en the follo wing inequalities are s atisfied:     1 + T 4 1 − T 4 − 2 T 2 (1 − T 4 ) cos κ     < 1 . (27) In addition, n otice that τ n = λ − 1 n, 0 τ 0 . Th is im p lies that the linear stabilit y of the normal iteration pro cedure is stable when λ n, 0 6 = 0 for all n . Th us, th is linear stabilit y is v alid only pr o vided κ 6 = k π 2 ( k = 0 , ± 1 , ± 2 , · · · ) . Ob viously , this n ecessary stabilit y condition is also in clud ed in the cond ition (27). As sho wn in Fig.3, parameters selected fr om the blac k regions violate the inequalit y (27). Those diamond-lik e u nstable regions p erio dically app ears in a p erio d 2 π with resp ect to the parameter κ . Figur e 3 ar ound her e. Ple ase find it at the end of the p ap er. The relation of the linear stabilit y an alysis in this sub section and the transmission exp osition in the subsection A is that the n on-transmitting regions in Figure 1 or Figure 2 are alwa ys con tained in the u n stable regions in Figure 3 (b). This also indicates that the transmission exp osition for solutions with the t yp e (23) to Eq.(19) is a more accurate de- scription of the linear stabilit y analysis for Eq.(19), which exhib its some c haotic d ynamics of th e Eq.(14) and hence those of the DHF (1). § 4 Conclusion W e p resen t a mec hanism f or disp laying the transmission prop erty of the discrete Heisen- b erg ferromagnetic spin c hain via a geometric approac h. More pr ecisely , we firs t trans form the DHF (1) in to the d iscrete n onlinear Schr¨ odinger-lik e equation (14) with the aid of the (discrete) Y ang-Mills theory . Then w e use the stationary v ersion of the discrete nonlinear Sc hr¨ od in ger-lik e equation (14) to stu d y th e transmission problem of the DHF (1). In this 11 pro cedure, we sho w that the determination of transmitting co efficien ts in the bac kw ard transmission problem is alw a ys b istable. Thus a defin ite algorithm and general sto c hastic algorithms are presen ted. Th e corresp onding tr ansmitting b eha viors in the parameters ( κ, T ) plane are sh o wn by Figure 1 and 2. A new inv arian t p erio dic phenomenon of the non-transmitting b eha vior for the DHF, with a large p robabilit y , is rev ealed b y an adop- tion of v arious sto c hastic algorithms. By th e w a y , an appr o ximate linear s tability analysis for th e stationary ve rsion of Eq.(19 ) is also give n for supp orting the d iscussion of th e transmission exp ositions. W e remark that though th e gauged equiv alent equation (14) with (13) lo oks muc h complicated and we cannot giv e a direct physical application at present stage from it, it is an effectiv e wa y to utilize it to study some dyn amical prop erties of the DHF (1). W e b eliev e that the discrete n onlinear Sc hr ¨ odinger-lik e equation (14) can help u s to reve al muc h more and deep er quantum c haotic prop erties of the DHF (1). Ac kno wledgmen ts This w ork was supp orted b y the National Natural S cience F oundation of China (Gran t Nos. 10531090 , 10501008 ) and STCS M. References [1] N. Papanicolaou, J. Phys. A: Ma th. Gen. 2 0 (19 87) 3637 . [2] M. Laksma nan, P hys. Lett. A 61 (1977 ) 53 . [3] L.A. T akhta jan, Phys. Lett. A 64 (1 977) 235 . [4] Y. Ishimori, J. Phys. So c. Ja pa n, 52 (19 82) 3417 . [5] K . Porsezian, M. Daniel and M. Lakshmanan, J. Math. Phys. 33 (1992) 1907. [6] M. Daniel and R. Am uda, Phys. Rev. B 5 3 (1996 ) R29 30. [7] Q . Ding, P hys. Lett. A 26 6 (2000 ), 14 6. [8] H.J. Mikesk a and M. Steiner , Adv. P hys. 44 (1991) 19 1. [9] V.E. Zakharov and L.A. T a kht a jan, Theor . Math. Phys. 38 (19 79) 17 . [10] L .D. F addeev a nd L.A. T akhta jan, Hamiltonian Methods in the Theor ey of Solitons, Springer - V erla g, Berlin, Heidebe rg 19 87. [11] Q . Ding and Z. Zhu, J. Phys. So c. Japa n 72 No.1 (2003 ), 49 . [12] Q . Ding, J.Phys. A: Math. Theo r. 40 (2007) 19 9 1. [13] F. Delyon, Y.E. L evy and B . Souilla r d, Phys. Rev. Lett. 57 (198 6) 2 010. [14] B .M. Herbs t, and M.J . Ablowitz, Phys. Rev. Lett. 6 2 (1989 ) 20 65. 12 [15] Yi W an a nd C.M. So ukoulis, Phys. Rev. A 41 ,80 0 (199 0) [16] D. Hennig, N.G. Sun, H. Ga briel and G.P . Tsiro nis, P hys. Rev. E 5 2 (1995 ) 25 5. [17] C . Flytzanis, in Nonline ar Phenomenon in Solids , e dited by A.F. V avrek (W orld Scientific, Singap ore, 19 8 5). [18] P .G. K evrekidis, K. Rasmussen and A.R. Bishop, Inter. J. Mo d. P hys. B 15 (200 1) 2 833. [19] C . Robinso n, “Dynamical Systems: Sta bilit y , Symbolic Dynamics, a nd Chaos,” 2nd E d., CRC Press, 19 98. [20] R. Grimshaw, “ Nonlinear O rdinary Differential Equations,” Bla ckw ell, O xford, 19 90. 13 (a) (b) (c) Fig. 1: The transmitting b ehavior in the ( κ, T ) parameters pla ne ( κ, T ) ∈ (1 , 3 . 5) × (0 . 5 , 2) (a) with N = 110 and (b) w ith N = 15 0 , and in an en- larged reg ion ( κ, T ) ∈ (1 . 3 , 1 . 7) × (1 , 1 . 4) (c) with N = 150. The b ounda ries of transmitting (white) and non-trans mitting (bla ck) reg ions exhibit fractal-like structure. . 14 1 2 3 4 5 6 1 1.1 1.2 1.3 1.4 κ T (a) 1 2 3 4 5 6 1 1.1 1.2 1.3 1.4 κ T (b) Fig.2. The p erio dic pheno menon o f the no n-transmitting b ehavior in ( κ, T ) plane by adopting of the normal distribution (a) and a Poisson distribution (b). Here the chain length N = 15 0. 15 (a) (b) Fig.3. The linear stability par ameters r egions (white for stable region a nd black for unstable region) when ( κ, T ) is in (0 , 13) × (0 , 2) (a) and in (1 , 3 . 5) × (0 , 2) (b), resp ectively . 16