Thermodynamics of antiferromagnetic alternating spin chains

Thermo dynamics of an tiferromagnetic alternating spin c hains G.A.P . Rib eiro ∗ and A. Kl ¨ ump er † Theoret isc he Physik, Bergische Univ er sit¨ at W upp ert a l, 4209 7 W upp erta l, Germa n y No v emb er 20, 20 18 Abstract W e consider inte grable quant um spin c h ains with alternating spins ( S 1 , S 2 ). W e deriv e a fin ite set of non-linear in tegral equ ations f or the thermo dyn amics of th ese mo dels by use of the quantum transf er m a- trix app roac h. Numerical solutions of the integral equations are pro- vided f or quan tities lik e sp ecific heat, magnetic su sceptibilit y and in the case S 1 = S 2 for the thermal Dru de weigh t. A t lo w temp eratures one class of m o dels sh o ws finite m agnetizat ion and the other class present s antiferromagnetic b eha viour . The thermal Drude weig h t b e- ha ves linearly on T at lo w temp eratures and is p rop ortional to the ∗ pav a n@ physik.uni-wuppertal.de † kluempe r@physik.uni-wupper tal.de cen tral c harge c of the system. Quite generally , w e observ e residu al en tropy f or S 1 6 = S 2 . P A CS n um b ers: 05.50+q, 02.30.IK, 05.70Jk Keyw ords: Bethe Ansatz, Thermo dynamics, Quan tum transfer matrix, Mixed spin chain 2 1 In tro duct ion In tegrable quan tum systems and their asso ciated classical vertex mo dels ha v e b een extensiv ely studied in the la st decades [1, 2]. A large part of these systems is exactly solv able b y Bethe ansatz tec hniques pro viding spectral data and in some cases also the eigen v ectors. After establishing the in tegrability and deriving the exact solution for the sp ectrum, the main questions one likes to answ er concern the ph ysical prop erties of the system in dep endence on temp erature, mag netic field etc. There are man y inv estigations o f in t egr a ble system in the thermo dynamical limit a t finite temp erature. In fa ct, w e hav e sev eral established routes to this goal. One may minimize the free energy functional in the com binato r ia l Thermo dynamical Bethe Ansatz approach ( TBA) [3, 4, 5], or one may apply algebraic and analytical means for the computat io n o f the partition function from the quan tum transfer matrix (QTM) [6, 7]. The TBA approac h is based o n the string hypothesis and yields an infinite set of non-linear integral equations (NLIE). How ev er, it is impractical to solv e the TBA equations n umerically due to the infinite num b er of equations and unkno wns. Therefore approximations are required in this approach. By means o f the quan tum t r ansfer matrix a ppro ac h, a finite set of NLIE can b e deriv ed exploiting a nalyticit y prop erties of the quantum transfer ma- trix. These equations ha v e b een sho wn to b e succes sful in the description of thermo dynamical prop erties in the complete temp erature range for man y imp ortant mo dels, lik e the Heisen b erg mo del [7, 8, 9 ] and its spin- S gen- eralization [10 ], the t − J mo del [11], the Hubbard mo del [12] and S U ( N ) in v ariant mo dels for N ≤ 4 [13]. 3 Nev ertheless, the standard construction of the quan tum transfer matrix assumes mo dels with isomorphic auxiliary and quantum spaces. Here w e a re concerned with extensions to more general mo dels with non-isomorphic a ux- iliary a nd quan t um spaces. Imp ortan t examples of suc h sys tems are mixed spin ch ains. These mixed c hains hav e b een extensiv ely studied for low and high temp eratures b y use of the TBA equations a nd finite size scaling for isotropic c hains [14, 1 5]. The dep endence on magnetic fields w as studied [16, 17, 18, 19] and more recen tly , also the anisotr o pic generalization w as considered [20]. Our aim is to prop ose a construction of the quan tum transfer matrix b y r eplacing the standard “rotation” of ve rtex configurations of Boltzmann w eights by conjugated represen tations, i.e. by the normal Boltzmann w eigh t shifted by the crossing parameter. Ha ving this in mind, w e can tac kle the more general situation where the auxiliary and quantum spaces are no t iso- morphic. As an application of this idea, w e study the generic ( S 1 , S 2 ) case of alternating spin c hains at finite temp erature. The paper is organized as f ollo ws. In section 2, w e outline the basic ingredien ts of the quantum transfer matr ix approa c h. In section 3, we define the alternating spin c hain and its prop erties. In section 4, we derive the set of non-linear integral equations. In section 5, w e presen t our nume rical findings for the solution of the NLIE. Section 6 is devoted to the calculation of the thermal D rude we igh t for the case S 1 = S 2 . O ur conclusions are give n in section 7. 4 2 Quan tum trans fer matrix W e are inte rested in the computation of the partition function Z = T r e − β H in the thermo dynamical limit, on the condition that H is an inte grable lo cal Hamiltonian deriv ed from some row -to-row transfer matrix. In general, transfer matrices can b e constructed a s o rdered pro ducts of man y differen t lo cal Boltzmann w eigh ts L A i ( λ ), where λ denotes the sp ectral parameter. These we igh ts can b e considered as matrices on the space A , usually called auxiliary space, whic h is related to the degrees of freedom o n the horizontal lines of a tw o dimensional vertex mo del. The matrix elemen ts of L A i ( λ ) are op erators acting non-tr ivially on the site i o f the quan t um space Q L i =1 V i of a c ha in of length L and are related to the degrees of freedom on v ertical lines. The pro duct of Boltzmann we igh ts T A ( λ ) = L A L ( λ ) L A L − 1 ( λ ) . . . L A 1 ( λ ) , (1) defines the mono dromy matrix T A ( λ ). Here w e allow ed for non-isomorphic spaces V i . This w a y , L A i ( λ ) – also called L -op erator s– ma y hav e differen t represen tations for the L man y quan tum spaces L A i ( λ ) = L ( α,β i ) A i ( λ ). The lab els f o r different represen tatio ns, α , β i , ma y take fo r instance inte ger v alues α, β i = 0 , . . . , L − 1 a nd L ( α,α ) A i ( λ ) denotes the isomorphic represen tation. Then the row-to-ro w transfer matrix is the trace o v er the a uxiliary space of the mono dromy matrix, T ( λ ) = T r A [ T A ( λ )] . (2) The transfer matrix constitutes a family of comm ut ing op erators [ T ( λ ) , T ( µ )] = 0, provide d there is an inv ertible R -matrix acting on the tensor pro duct 5 A ⊗ A , suc h that R ( α ) ( λ − µ ) L ( α,β i ) A i ( λ ) ⊗ L ( α,β i ) A i ( µ ) = L ( α,β i ) A i ( µ ) ⊗ L ( α,β i ) A i ( λ ) R ( α ) ( λ − µ ) . (3) In order to ha v e a n asso ciativ e algebra, the R -matrix is required to satisfy the Y ang-Baxter equation R ( α ) 12 ( λ ) R ( α ) 23 ( λ + µ ) R ( α ) 12 ( µ ) = R ( α ) 23 ( µ ) R ( α ) 12 ( λ + µ ) R ( α ) 23 ( λ ) . (4) The simplest solution of (3) o ccurs when a uxiliary a nd quantum spaces V i are isomorphic implying that L ( α,α ) 12 ( λ ) = P 12 R ( α ) 12 ( λ ), where P 12 is the p erm uta t io n op erator. The conserv ed charges are obtained through the deriv ativ es o f the lo ga- rithm of the transfer matrix J ( n ) = ∂ n ∂ λ n ln [ T ( λ )]    λ =0 , (5) and the Hamiltonian correspo nds to the first deriv ativ e, H = J (1) . Therefore, w e can relate the tra nsfer matr ix a nd the Hamiltonian in the following w ay T ( λ ) = T (0) e λ H + O ( λ 2 ) , (6) where T (0) pla ys the role of a kind of righ t m ultiple-step shift op erator [14] for a g eneral distribution of L -o p erators L ( α,β i ) A i ( λ ). Let us consider that in addition to relation (3) t he L - op erators satisfy the follo wing symmetry prop erties Unitarit y: L ( α,β ) 12 ( λ ) L ( α,β ) 12 ( − λ ) = ζ α,β ( λ )Id 1 ⊗ Id 2 , (7) Time rev ersal: L ( α,β ) 12 ( λ ) t 1 = L ( α,β ) 12 ( λ ) t 2 , (8) Crossing: L ( α,β ) 12 ( λ ) = ς α,β ( λ ) M 1 L ( α,β ) 12 ( − λ − ρ ) t 2 M − 1 1 , (9) 6 where ζ α,β ( λ ) and ς α,β ( λ ) are scalar functions and ρ is the crossing parameter. Here Id i and t i denote the iden tity matrix and transp osition on the i - th space, M 1 = M ⊗ Id 2 where M is some scalar matrix. No w, we can define an adj o in t t r ansfer matrix T ( λ ) as follow s T ( λ ) = L Y i =1 ς α,β i ( λ ) T r A h L ( α,β L ) A L ( − λ − ρ ) L ( α,β L − 1 ) A L − 1 ( − λ − ρ ) . . . L ( α,β 1 ) A 1 ( − λ − ρ ) i , (10) and b y using the pro p erties (8-9) w e can rewrite the tra nsfer matrix T ( λ ) suc h tha t, T ( λ ) = T r A h L ( α,β 1 ) A 1 ( λ ) . . . L ( α,β L − 1 ) A L − 1 ( λ ) L ( α,β L ) A L ( λ ) i . (11) Here w e can see that, due to unita rit y (7), the logar it hmic deriv ativ e results in the same Hamiltonian H = H and T (0) corresp onds to the left m ultiple- step shift op erator, suc h that T (0) T (0) = N Id where N = Q L i =1 ζ α,β i (0). In analogy to (6), w e can write the transfer matrix T ( λ ) as T ( λ ) = T (0) e λ H + O ( λ 2 ) . (12) Using ( 6 ) and (12) w e can rewrite the partition function Z in terms of the transfer matrices T ( λ ) and T ( λ ) by considering the T rotter limit, Z = lim N →∞ T r h ( e − 2 β N H ) N/ 2 i , (13) = lim N →∞ T r h  T ( − τ ) T ( − τ )  N/ 2 i 1 N N/ 2 , τ := β N . (14) The partit io n function (14) can b e related to a staggered v ertex mo del with a lternating ro ws T and T . In this case we need to know all t he eigen- v alues of these tw o tra nsfer matrices to obtain the partition function in a 7 closed form. This is due to the fa ct that the eigen v alues of b oth tra nsfer matrices dep end on the length of the quan tum c hain L and in par ticular on the T rotter n umber N , suc h that for N → ∞ all ga ps close. Ho w ever, w e can circum v ent this problem b y rewriting (14) in terms of the column-to-column transfer matr ix describing transfer in chain direction and hence is called the quan tum transfer matrix T QT M i ( x ) ( ς α,β i ( − (i x + τ ))) N/ 2 = T r V i [ L ( β i ,α ) V i N (i x + τ − ρ ) L ( β i ,α ) V i N − 1 (i x − τ ) . . . L ( β i ,α ) V i 2 (i x + τ − ρ ) L ( β i ,α ) V i 1 (i x − τ ) ] . (15) Eac h of these o b jects has a well defined la rgest eigenv alue separated b y a gap from the r est of the sp ectrum, ev en in the limit N → ∞ . Therefore, o nly the la rgest eigen v alue is required for the computation of the partition func- tion. Here x is the sp ectral par ameter asso ciated with the v ertical line ensur- ing the existence o f a comm uting f amily of matrices, h T QT M i ( x ) , T QT M i ( x ′ ) i = 0. How ev er, of direct physic al relev ance is x = 0 for obtaining t he par tition function, Z = lim N →∞ T r " L Y i =1 T QT M i (0) # 1 N N/ 2 . (16 ) Next, w e address the iden tification of the largest eigen v alue of the pro duct of the quantum transfer matrices T QT M i ( x ). In general, the determination of the largest eigen v alue of the pro duct of matrices Q L i =1 T QT M i ( x ) w ould require the know ledge of all the eigen v alues of all transfer matrices T QT M i ( x ), whic h could turn out to b e a more in volv ed problem than the staggered mo del men tioned ab ov e. Nev ertheless, this problem can b e o v ercome under certain conditions. F or instance , for the case of mixed spin c hains all of the transfer matrices 8 comm ute according to the Y ang -Baxter equation and the largest eigenv alues of the individual transfer matrices corresp ond to the same eigen vec tor. This implies that the largest eigen v alue of the pro duct o f L differen t transfer matrices is nothing than the pro duct of the largest eigenv alues of the quan tum transfer matrices. In t his w o r k, w e will restrict to this sp ecific case. Here w e are in terested in the free energy and its deriv a tiv es, so w e ha ve to consider the logarithm o f the pa r tition function in the infinite length limit. As the eigen v alues Λ QT M i ( x ) dep end only on the T rotter n um b er, w e can first tak e the infinite length limit and lat er the infinite T rotter num b er limit, whic h reads f = − 1 β lim L,N →∞ 1 L ln [ Z ] , (17) = − 1 β lim N ,L →∞ 1 L L X i =1 ln h Λ QT M i,max (0) i + 1 β lim N ,L →∞ 1 L ln  N N/ 2  . (18) Before closing this section, w e would lik e to men tion that the pr o p erties (7-9) are also satisfied b y man y isomorphic self-crossed mo dels [21]. F or the S U ( N ) case with N > 2, the prop ert y (9) reduces to the standard “rotation” of the v ertex configuratio n of the Boltzmann w eights. 3 Alternatin g spin c hains In the previous section, w e used unitarity , time rev ersal a nd crossing prop- erties to construct the quan t um tra nsfer matrix considering general repre- sen tations of L ( α,β i ) A i ( λ ). F r o m now on, w e consider (for an ev en num b er of lattice sites L ) the alternation of t w o differen t represen tatio ns of the group S U (2) with spin S 1 at o dd sites and spin S 2 at eve n sites , i.e. β 2 i − 1 = S 1 9 and β 2 i = S 2 . In o r der t o hav e a Hamilto nian with lo cal interactions we fix α to b e iden t ical to the spin S 1 represen tation (equiv alen tly we could hav e c hosen S 2 ). The mono dromy matrix (1) b ecomes T ( S 1 ,S 2 ) A ( λ ) = L ( S 1 ,S 2 ) A L ( λ ) L ( S 1 ,S 1 ) A L − 1 ( λ ) . . . L ( S 1 ,S 2 ) A 2 ( λ ) L ( S 1 ,S 1 ) A 1 ( λ ) , (19) with the auxiliary space A ≡ C 2 S 1 +1 , and L ( S 1 ,S 2 ) A i ( λ ) resp. L ( S 1 ,S 1 ) A i ( λ ) are the L -op erato rs with spin S 1 represen tation in the auxiliary space a nd S 2 resp. S 1 in the quan tum space. The ab ov e S U (2) inv ariant L -op erators can b e obtained through the fu- sion pro cess [22]. Its explicit form con ven ien tly no r ma lized is giv en b y L ( S 1 ,S 2 ) 12 ( λ ) = S 1 + S 2 X l = | S 1 − S 2 | f l ( λ ) ˇ P l , (20) where 1 f l ( λ ) = Q S 1 + S 2 j = l +1  λ − j λ + j  Q ∗ 2 S 1 j =1 ( λ + S 2 − S 1 + j ) and ˇ P l is the pro jector on to the S U (2) l in the Clebsc h-Gordon decomposition S U (2) S 1 ⊗ S U (2 ) S 2 . This op erator is represen ted b y ˇ P l = S 1 + S 2 Y k = | S 1 − S 2 | k 6 = l ~ S 1 ⊗ ~ S 2 − x k x l − x k , (21) with x l = 1 2 [ l ( l + 1) − S 1 ( S 1 + 1) − S 2 ( S 2 + 1) ] and the S U (2) generators ~ S a = ( ˆ S x a , ˆ S y a , ˆ S z a ) for a = 1 , 2. The op erator (20) is a solution of (3) with the following R -matrix R ( S 1 ) 12 ( λ ) = P 12 L ( S 1 ,S 1 ) 12 ( λ ) . (22) 1 The symbol ∗ sha ll remind tha t the p ossibility j = S 1 − S 2 is excluded thro ug hout this work. 10 It satisfies the properties (7-9) with scalar functions giv en by ζ S 1 ,S 2 ( λ ) = Q 2 S 1 j =1 (( S 2 − S 1 + j ) 2 − λ 2 ) and ς S 1 ,S 2 ( λ ) = ( − 1) 2 S 1 and crossing parameter ρ = 1. The matrix M is a n an ti-diagonal matrix whose non-zero elemen ts are M i,j = − ( − 1) i δ i, 2 S 1 +2 − j . The Hamiltonia n asso ciated to the transfer matrix T ( λ ) = T r A h T ( S 1 ,S 2 ) A ( λ ) i has terms with tw o and three site in teractions. Its generic expression is g iv en b y H ( S 1 ,S 2 ) = X ev en i h L ( S 1 ,S 2 ) i − 1 ,i (0) i − 1 ∂ ∂ λ L ( S 1 ,S 2 ) i − 1 ,i ( λ )    λ =0 (23) + X odd i h L ( S 1 ,S 2 ) i − 2 ,i − 1 (0) i − 1 h L ( S 1 ,S 1 ) i − 2 ,i (0) i − 1 ∂ ∂ λ L ( S 1 ,S 1 ) i − 2 ,i ( λ )    λ =0 L ( S 1 ,S 2 ) i − 2 ,i − 1 (0) , where p erio dic b oundary conditions are assumed. F or illustrat ion, the Hamil- tonian for case S 1 = 1 / 2, S 2 = S is giv en explicitly by [15] H ( 1 2 ,S ) = 1 2  1 S + 1 2  2 h X ev en i  ~ σ i − 1 · ~ S i + ~ S i · ~ σ i +1 + n ~ σ i − 1 · ~ S i , ~ S i · ~ σ i +1 o +  1 4 − S ( S + 1 )  X ev en i ~ σ i − 1 · ~ σ i +1 i + L 4  1 + 1 ( S + 1 2 ) 2  . (24) One of the consequence s of the alternation of tw o differen t spins is that w e hav e t w o quantum transfer matrices to w ork with. W e denote them by T ( S 1 ) ( x ) and T ( S 2 ) ( x ), suc h as T ( S a ) ( x ) := T QT M a ( x ) = T r V a [ L ( S a ,S 1 ) V a N (i x + τ − ρ ) L ( S a ,S 1 ) V a N − 1 (i x − τ ) . . . L ( S a ,S 1 ) V a 2 (i x + τ − ρ ) L ( S a ,S 1 ) V a 1 (i x − τ ) ] , (25) where the v ertical spaces are V a ≡ C 2 S a +1 and a = 1 , 2. The transfer matrices (25) for a = 1 , 2 commu te due to the Y ang-Baxter relation [23]. Therefore, t hey can b e diagonalized sim ulta neously . It can also 11 b e deduced from [23] that t heir largest eigenv alues corresp o nd to the same eigenstate. Hence the largest eigen v alue of t he pro duct T ( S 1 ) ( x ) T ( S 2 ) ( x ) is the pro duct of the largest eigen v alues of T ( S 1 ) ( x ) and T ( S 2 ) ( x ). F or the analysis of the sp ectra w e use the fusion hierarch y for t he quantum transfer matrix T ( j ) ( x ), in analogy to the fusion of L -op erato r s. The algebraic relations read (see e.g. [10]) T ( j ) ( x ) T ( 1 2 ) ( x + i( j + 1 2 )) = a j ( x ) T ( j + 1 2 ) ( x + i 2 ) + a j +1 ( x ) T ( j − 1 2 ) ( x − i 2 ) , T (0) ( x ) = a 0 ( x )Id , j = 1 2 , 1 , 3 2 , . . . (26) where a j ( x ) = Q 2 S 1 l =1 φ + ( x + i( j − S 1 + l − 1)) φ − ( x + i( j − S 1 + l )) and φ ± ( x ) = ( x ± i τ ) N/ 2 . F rom the fusion hierarch y with bilinear and linear expressions in T (2 6), one can o btain another set of functional relations [2 4], usually called T - system, with exclusiv ely bilinear expressions T ( j ) ( x + i 2 ) T ( j ) ( x − i 2 ) = T ( j − 1 2 ) ( x ) T ( j + 1 2 ) ( x ) + f j ( x ) Id , (27) where f j ( x ) = Q 2 S 1 l =1 φ + ( x − i( j − S 1 + l + 1 2 )) φ − ( x − i( j − S 1 + l − 1 2 )) φ + ( x + i( j − S 1 + l − 1 2 )) φ − ( x + i( j − S 1 + l + 1 2 )) for an y j in teger or semi-in teger. Equally imp ortant is a set of functional relations referred to as the Y - system, whic h is a conseque nce of (27). It is written as y ( j ) ( x + i 2 ) y ( j ) ( x − i 2 ) = Y ( j − 1 2 ) ( x ) Y ( j + 1 2 ) ( x ) , (28) where y ( j ) ( x ) = T ( j − 1 2 ) ( x ) T ( j + 1 2 ) ( x ) f j ( x ) and Y ( j ) ( x ) = 1 + y ( j ) ( x ). Lastly , we introduce a Zeeman term e H = H − h ˆ S z . This term represen ts the coupling of the magnetic field h to the spin ˆ S z = P L i =1 odd i ˆ S z 1 ,i + P L i =1 ev en i ˆ S z 2 ,i . 12 It can b e in tro duced inside the trace of the partitio n function such a s, Z = lim N →∞ T r h  T ( − τ ) T ( − τ )  N/ 2 e β h ˆ S z i 1 N N/ 2 . (29) Alternativ ely , it can b e considered as a diagonal b oundary term on the v er- tical lines a long a horizontal seam. This redefines only trivially the quan tum transfer matrix T ( S a ) ( x ) = T r V a [ G a L ( S a ,S 1 ) V a N (i x + τ − ρ ) L ( S a ,S 1 ) V a N − 1 (i x − τ ) . . . L ( S a ,S 1 ) V a 2 (i x + τ − ρ ) L ( S a ,S 1 ) V a 1 (i x − τ ) ] , (30) where G a is a diagonal mat rix whose non-zero elemen ts are ( G a ) i,i = e β h ( S a +1 − i ) . The eigenv alues Λ ( j ) ( x ) asso ciated to T ( j ) ( x ) also satisfy the functional relations (26-28). This is due to the comm utativity prop ert y among differen t T ( j ) ( x ). This w ay , w e obtain the eigen v alues at an y fusion lev el in terms of the first lev el eigenv alue throug h the iteration of the relations (26) and (27). Alternatively , w e can pro ceed along the same lines as [25] applying the algebraic Bethe ansatz to the case of t wisted b oundary conditions. In b oth cases w e end up with the eigenv alues of the quantum transfer matrix (30), Λ ( j ) ( x ) = 2 j + 1 X m =1 λ ( j,S 1 ) m ( x ) , (31) λ ( j ) m ( x ) = e β h ( j +1 − m ) t ( j ) + ,m ( x ) t ( j ) − ,m ( x +i) Q ( x − i( 1 2 + j )) Q ( x + i( 1 2 + j )) Q ( x − i( 3 2 + j − m )) Q ( x − i( 1 2 + j − m )) , (32) where t ( j ) ± ,m ( x ) = j Y l = j − m +2 φ ± ( x − i( l − S 1 )) φ ± ( x − i( l + S 1 )) 2 S 1 Y l =1 ∗ φ ± ( x − i( j − S 1 + l ) ) and Q ( x ) = 13 Q n l =1 ( x − x l ). The cor r esp o nding Bethe ansatz equations can b e written as e β h φ + ( x l − i( S 1 + 1 2 )) φ − ( x l − i( S 1 − 1 2 )) φ − ( x l + i( S 1 + 1 2 )) φ + ( x l + i( S 1 − 1 2 )) = n Y j =1 j 6 = l x l − x j − i x l − x j + i . (33) According to t he previous section, w e o nly need to know the largest eigen- v alue in the limit N → ∞ to describe the thermodynamics of the one di- mensional quan t um mo del. Then for instance by n umerical a na lysis of the Bethe ansatz equation (33) for small N w e see that the largest eigen v alue lies in the sector n = S 1 N . Ho w eve r, t he limit N → ∞ cannot b e considered n umerically . So, w e need to enco de the Bethe ansatz ro ots in suc h a wa y that the free energy can b e ev aluated indep enden tly of the exact knowle dge of the individual ro o t s. One p ossible wa y is to define a set of suitable auxiliary functions de- p ending on the Bethe ansatz ro ots. Then by exploiting the ab ov e and fur- ther functional relations w e eliminate t he explicit dep endence on the ro ots. Therefore the Bethe ansatz ro ots for finite N (including the limit N → ∞ ) b ecome enco ded in a finite set of a uxiliary functions satisfying certain non- linear in tegra l equations. Suc h an analysis w as already done for many cases, for instance for the spin-1 / 2 Heisen b erg chain [7, 8, 9] and its higher spin extensions [10]. In the latter case, the auxiliary functions w ere ta k en a s a subset o f the y -functions complemen ted b y tw o “ no vel” functions whic h reduce the infinitely man y functional relatio ns (28) to finitely man y . This is the starting p oint o f the next section. 14 4 Non-linear in tegral equation s In this section, w e in tro duce a suitable set of auxiliary f unctions and explore its analyticit y prop erties t o obt a in a finite set of non- linear in tegra l equations. These auxiliary functions turn out to describ e the largest eigen v alue of (30) and consequen tly the free energy (18) at finite temp erature. Sp ecifically , we need to define 2 s + 1 auxiliary functions, where s = max( S 1 , S 2 ). W e will pro ceed a lo ng the lines of [1 0] and take as the first 2 s − 1 auxiliary functions the y -functions y ( j ) ( x ) = Λ ( j − 1 2 ) ( x )Λ ( j + 1 2 ) ( x ) f j ( x ) , j = 1 2 , . . . , s − 1 2 . (34) The tw o remaining f unctions are defined as b ( x ) = λ ( s ) 1 ( x + i 2 ) + · · · + λ ( s ) 2 s ( x + i 2 ) λ ( s ) 2 s +1 ( x + i 2 ) , (35) ¯ b ( x ) = λ ( s ) 2 ( x − i 2 ) + · · · + λ ( s ) 2 s +1 ( x − i 2 ) λ ( s ) 1 ( x − i 2 ) . (36) In addition to this, w e in tro duce a shorthand nota tion for simply related functions B ( x ) := 1 + b ( x ), ¯ B ( x ) := 1 + ¯ b ( x ) and Y ( j ) ( x ) := 1 + y ( j ) ( x ) for j = 1 2 , . . . , s − 1 2 . In confo rmit y with the previous definition, we note that B ( x ) = Λ ( s ) ( x + i 2 ) λ ( s ) 2 s +1 ( x + i 2 ) and ¯ B ( x ) = Λ ( s ) ( x − i 2 ) λ ( s ) 1 ( x − i 2 ) with pro duct B ( x ) ¯ B ( x ) = Y ( s ) ( x ). This implies for the first (2 s − 1) functional relations (28) y ( j ) ( x + i 2 ) y ( j ) ( x − i 2 ) = Y ( j − 1 2 ) ( x ) Y ( j + 1 2 ) ( x ) for j = 1 2 , 1 , . . . , s − 1 , (37) y ( s − 1 2 ) ( x + i 2 ) y ( s − 1 2 ) ( x − i 2 ) = Y ( s − 1) ( x ) B ( x ) ¯ B ( x ) . (38) 15 W e can write b ( x ) , ¯ b ( x ) , B ( x ) and ¯ B ( x ) explicitly using (32) suc h that b ( x ) = Q ( x + i( s + 1)) Q ( x − i s ) e β h ( s + 1 2 ) Λ ( s − 1 2 ) ( x ) Q 2 S 1 l =1 φ + ( x + i( s − S 1 + l − 1 2 )) φ − ( x + i( s − S 1 + l + 1 2 )) , (39) ¯ b ( x ) = Q ( x − i( s + 1 )) Q ( x + i s ) e − β h ( s + 1 2 ) Λ ( s − 1 2 ) ( x ) Q 2 S 1 l =1 φ + ( x − i( s − S 1 + l + 1 2 )) φ − ( x − i( s − S 1 + l − 1 2 )) , (40) B ( x ) = Q ( x + i s ) Q ( x − i s ) e β hs Λ ( s ) ( x + i 2 ) Q 2 S 1 l =1 φ + ( x + i( s − S 1 + l − 1 2 )) φ − ( x + i( s − S 1 + l + 1 2 )) , (41) ¯ B ( x ) = Q ( x − i s ) Q ( x + i s ) e − β hs Λ ( s ) ( x − i 2 ) Q 2 S 1 l =1 φ + ( x − i( s − S 1 + l + 1 2 )) φ − ( x − i( s − S 1 + l − 1 2 )) . (42) In this w ay , it is eviden t tha t b ( x ), ¯ b ( x ) are related to Λ ( s − 1 2 ) ( x ). Moreo ve r, Λ ( s − 1 2 ) ( x ) is related to Y ( s − 1 2 ) ( x ) through the definition of y - function. This relation can b e written as Λ ( s − 1 2 ) ( x + i 2 )Λ ( s − 1 2 ) ( x − i 2 ) = f s − 1 2 ( x ) Y ( s − 1 2 ) ( x ) . (43) A t this p o in t, we hav e a common set of functions whic h still dep end on the Bethe ansatz ro ots and whose limit N → ∞ is still to b e p erformed. Ho we v er, this dep endence as w ell as the limit can b e w orked out easily in F ourier space. In order to calculate the F ourier transform, we exploit the analyticity prop erties of the eigen v alue of the quan tum transfer matrix a nd the auxil- iary functions. F urthermore, these functions should b e non-zero and hav e constan t asymptotics in a strip around the real axis. This allows us to apply 16 the F ourier tra nsform to the logarithmic deriv ativ e of the auxiliary f unctions, ˆ f ( k ) = Z ∞ −∞ d dx [ln f ( x )] e − i k x dx 2 π . (44) In the cases k < 0 and k > 0, w e ha v e c hosen a closed con tour a b ov e and b elo w the real axis, respectiv ely . F or this reason, it is of fundamental imp ortance to analyze the structure o f t he zeros of the auxiliary functions. In particular, the zeros and p oles of the auxiliary functions (34-36) origi- nate fr o m the zeros of Q ( x ) and Λ ( j ) ( x ) fo r j = 1 2 , . . . , s b esides those of the φ ± ( x ) functions. The refore, we hav e t o analyze t he qualitative distribution of the Bethe ansatz ro ots as w ell as the zeros of t he eigenv alue functions Λ ( j ) ( x ). It is w ell known that Bethe a nsatz ro o ts form 2 S 1 -strings in the particle sector n = S 1 N . Thes e ro ots hav e imaginary parts placed appro ximately at ( S 1 + 1 2 − l ) for l = 1 , . . . , 2 S 1 [23]. Concerning the zeros of Λ ( j ) ( x ) for j = 1 2 , . . . , s , we hav e verified n umerically that their imaginary par t s are placed at ± ( j − S 1 + l ) f o r l = 1 , . . . , 2 S 1 and l 6 = S 1 − j . By direct insp ection of (3 4,37-43), w e note that almost all auxiliary func- tions are free of zeros and p oles in a strip containing − 1 / 2 ≤ ℑ ( x ) ≤ 1 / 2 . The exce ptions ar e y ( S 1 ) ( x ) for S 1 < S 2 and b ( x ) , ¯ b ( x ) for S 1 ≥ S 2 , which should b e treated separately . This wa y , the p osition of the zeros and p oles of the auxiliary functions dep end on the relative magnitude of S 1 and S 2 . So, w e hav e to split our analysis in three parts: S 1 < S 2 , S 1 = S 2 and S 1 > S 2 . 17 4.1 S 1 < S 2 In this case, we hav e s = S 2 in the previous definition. In or der to deal with the problem in v olving the f unction y ( S 1 ) ( x ), w e define a related function for whic h the pro blematic zeros and p oles at x = ± i / 2 are cancelled, ˜ y ( S 1 ) ( x ) = φ + ( x + i 2 ) φ − ( x − i 2 ) φ − ( x + i 2 ) φ + ( x − i 2 ) y ( S 1 ) ( x ) . (45) Consequen tly , the 2 S 1 -th equation in (37) b ecomes ˜ y ( S 1 ) ( x + i 2 ) ˜ y ( S 1 ) ( x − i 2 ) = φ − ( x − i) φ + ( x + i) φ + ( x − i) φ − ( x + i) Y ( S 1 − 1 2 ) ( x ) Y ( S 1 + 1 2 ) ( x ) , (46) and the functions ˜ y ( S 1 ) ( x ± i 2 ) can b e transformed as usual according to ( 4 4). On the other hand, w e can apply the F ourier t ransform to the equation (4 5 ), once it do es not hav e zeros and p oles on the real axis. Thus we are able to establish a relation b etw een y ( S 1 ) and ˜ y ( S 1 ) in F o urier space, ˆ ˜ y ( S 1 ) ( k ) = i N sinh [ k β / N ] e −| k | / 2 + ˆ y ( S 1 ) ( k ) . (47) No w, applying (44) to the functional relat io ns ( 3 7-43) and (46) we ob- tain after a long but straigh t forw ar d calculation a set of a lgebraic relations in F ourier space. These relation a re giv en in terms o f the transformed aux- iliary functions ˆ y ( j ) ( k ), ˆ b ( k ), ˆ ¯ b ( k ), ˆ Y ( j ) ( k ), ˆ B ( k ), ˆ ¯ B ( k ) and the unkno wns ˆ Λ ( S 2 − 1 2 ) ( k ), ˆ Λ ( S 2 ) ( k ) and ˆ Q ( k ). W e can eliminate the unkno wns after some 18 algebraic manipulation. Finally , using (47) we obtain                  ˆ y ( 1 2 ) ( k ) . . . ˆ y ( S 1 ) ( k ) . . . ˆ y ( S 2 − 1 2 ) ( k ) ˆ b ( k ) ˆ ¯ b ( k )                  =                  0 . . . ˆ d ( k ) . . . 0 0 0                  + ˆ K ( k )                  ˆ Y ( 1 2 ) ( k ) . . . ˆ Y ( S 1 ) ( k ) . . . ˆ Y ( S 2 − 1 2 ) ( k ) ˆ B ( k ) ˆ ¯ B ( k )                  , (48) where the k ernel ˆ K ( k ) is a (2 S 2 + 1) × (2 S 2 + 1) matrix given by ˆ K ( k ) =                  0 ˆ K ( k ) 0 · · · 0 0 0 0 ˆ K ( k ) 0 ˆ K ( k ) . . . . . . . . . . . . 0 ˆ K ( k ) 0 0 0 0 . . . 0 ˆ K ( k ) 0 0 0 0 · · · 0 ˆ K ( k ) 0 ˆ K ( k ) ˆ K ( k ) 0 0 · · · 0 0 ˆ K ( k ) ˆ F ( k ) − e − k ˆ F ( k ) 0 0 · · · 0 0 ˆ K ( k ) − e k ˆ F ( k ) ˆ F ( k )                  , (49) with ˆ K ( k ) = 1 2 cosh [ k / 2] , ˆ F ( k ) = e −| k | / 2 2 cosh [ k / 2] and ˆ d ( k ) = − i N sinh [ k β / N ] 2 cosh [ k / 2] . As the T rotter n um b er N app ears only in ˆ d ( k ), w e can t a k e the limit N → ∞ straig htforw ardly , ˆ d ( k ) = − i 2 cosh [ k / 2] lim N →∞ N sinh [ k β / N ] = − i k β 2 cosh [ k / 2] . (50) The in ve rse F ourier transform has been applied to (48) follow ed b y an 19 in tegrat io n o v er x , resulting in                  ln y ( 1 2 ) ( x ) . . . ln y ( S 1 ) ( x ) . . . ln y ( S 2 − 1 2 ) ( x ) ln b ( x ) ln ¯ b ( x )                  =                  0 . . . − β d ( x ) . . . 0 β h 2 − β h 2                  + K ∗                  ln Y ( 1 2 ) ( x ) . . . ln Y ( S 1 ) ( x ) . . . ln Y ( S 2 − 1 2 ) ( x ) ln B ( x ) ln ¯ B ( x )                  , (51) where d ( x ) = π cosh [ π x ] and t he sym b ol ∗ denotes the con volution f ∗ g ( x ) = R ∞ −∞ f ( x − y ) g ( y ) dy . The in tegration constan ts ± β h/ 2 w ere determined in the asymptotic limit | x | → ∞ . The k ernel matrix is giv en explicitly b y K ( x ) =                  0 K ( x ) 0 · · · 0 0 0 0 K ( x ) 0 K ( x ) . . . . . . . . . . . . 0 K ( x ) 0 0 0 0 . . . 0 K ( x ) 0 0 0 0 · · · 0 K ( x ) 0 K ( x ) K ( x ) 0 0 · · · 0 0 K ( x ) F ( x ) − F ( x + i) 0 0 · · · 0 0 K ( x ) − F ( x − i) F ( x )                  , (52) where K ( x ) = π cosh [ πx ] and F ( x ) = R ∞ −∞ e −| k | / 2+i kx 2 cosh [ k / 2] dk . No w, we ha ve to deriv e an expression for the eigenv alue Λ ( S 2 ) ( x ) in terms of the auxiliary functions. It is con venie n t to define a new function Λ ( S 2 ) ( x ) = Λ ( S 2 ) ( x ) Q 2 S 1 l =1 φ + ( x − i( S 2 − S 1 + l )) φ − ( x + i( S 2 − S 1 + l )) , (5 3 ) 20 whic h has constant asymptotics. F or x = 0 a nd finite N , w e hav e ln Λ ( S 2 ) (0) = ln Λ ( S 2 ) (0) + P 2 S 1 l =1 ln h 1 − β S 2 − S 1 + l 1 N i N + 2 L ln  N N/ 2  , where we ha ve used the fact that N = Q 2 S 1 l =1 ( S 2 − S 1 + l ) L . Using the F ourier transformed v ersion of (41-4 2,53), w e obtain ˆ Λ ( S 2 ) ( k ) = i k β e −| k | ( S 2 − S 1 − 1 2 ) 2 cosh [ k / 2] 2 S 1 X l =1 e −| k | l + ˆ K ( k ) h ˆ B ( k ) + ˆ ¯ B ( k ) i . (54) Pro ceeding as b efore, w e apply t he in ve rse F ourier transform f ollo w ed b y an integration ov er x and the determination of the in tegra t io n constan t. In this w ay , we obtain ln Λ ( S 2 ) ( x ) = β ǫ ( S 2 ,S 1 ) ( x ) +  K ∗ ln B ¯ B  ( x ) , (55) where ǫ ( S 2 ,S 1 ) ( x ) is giv en by ǫ ( S 2 ,S 1 ) ( x ) = 2 S 1 X l =1 Z ∞ −∞ e −| k | ( S 2 − S 1 + l − 1 2 ) 2 cosh [ k / 2] e i k x dk . (56) A t the p oin t x = 0, w e can rewrite this in tegral in terms of the Euler psi function, ǫ ( S 2 ,S 1 ) (0) = ψ  S 2 + S 1 + 1 2  − ψ  S 2 − S 1 + 1 2  . (57) The con tribution of t he quan tum transfer matrix T ( S 2 ) (0) (30) to the free energy is g iv en by (18) f ( S 2 ,S 1 ) = − 1 2 β lim N →∞ ln Λ ( S 2 ) (0) + 1 β lim N ,L →∞ 1 L ln  N N/ 2  , (58) = − 1 2 β lim N →∞ ln Λ ( S 2 ) (0) + 1 2 2 S 1 X l =1 1 S 2 − S 1 + l . (59) 21 Therefore, w e can write f ( S 2 ,S 1 ) explicitly as f ( S 2 ,S 1 ) = 1 2 " 2 S 1 X l =1 1 S 2 − S 1 + l − ψ  S 2 + S 1 + 1 2  + ψ  S 2 − S 1 + 1 2  # − 1 2 β  K ∗ ln B ¯ B  (0) . (60) 4.2 S 1 = S 2 In this case, w e note that b ( x ) and ¯ b ( x ) hav e zeros a t x = ± i / 2 whic h are presen ting some subtleties. These zeros originate from t he factor Λ ( S 1 − 1 2 ) and in principle do not presen t any problems for the computation of the F o urier transform of the logarithmic deriv ativ e o f (39-40). Th e problem arises in the F o urier tr ansform of (43), whic h is required to eliminate the unkno wn function Λ ( S 1 − 1 2 ) . Hence, w e define a new function e Λ ( S 1 − 1 2 ) ( x ) = Λ ( S 1 − 1 2 ) ( x ) φ + ( x − i / 2) φ − ( x +i / 2) , which do es not hav e an y zeros at x = ± i / 2. W e apply (4 4) to the functional relations (37-43) with e Λ instead of Λ. Then w e eliminate the unkno wns ˆ Λ ( S 1 − 1 2 ) ( k ) and ˆ Q ( k ) and finally we o btain            ˆ y ( 1 2 ) ( k ) . . . ˆ y ( S 1 − 1 2 ) ( k ) ˆ b ( k ) ˆ ¯ b ( k )            =            0 . . . 0 ˆ d ( k ) ˆ d ( k )            + ˆ K ( k )            ˆ Y ( 1 2 ) ( k ) . . . ˆ Y ( S 1 − 1 2 ) ( k ) ˆ B ( k ) ˆ ¯ B ( k )            , (61) where the k ernel ˆ K ( k ) with the same structure a s (4 9 ), is a (2 S 1 + 1) × (2 S 1 + 1) matrix. Applying the in v erse F ourier transform to ( 61) follow ed b y a n integration 22 o ver x , results in            ln y ( 1 2 ) ( x ) . . . ln y ( S 1 − 1 2 ) ( x ) ln b ( x ) ln ¯ b ( x )            =            0 . . . 0 − β d ( x ) + β h 2 − β d ( x ) − β h 2            + K ∗            ln Y ( 1 2 ) ( x ) . . . ln Y ( S 1 − 1 2 ) ( x ) ln B ( x ) ln ¯ B ( x )            , (62) where the (2 S 1 + 1) × (2 S 1 + 1) ke rnel matrix is give n by (52). Finally , t he larg est eigenv alue Λ ( S 1 ) (0) of the quan tum transfer matrix T ( S 1 ) (0) (30 ) can b e written in terms o f the auxiliary functions in analogy to the previous case. W e just hav e to set S 2 = S 1 in a ll expressions (53- 60) and obtain, ln Λ ( S 1 ) (0) = β  ψ  2 S 1 + 1 2  − ψ  1 2  +  K ∗ ln B ¯ B  (0) . (63) Its con tribution t o the free energy is giv en by f ( S 1 ,S 1 ) = 1 2 " 2 S 1 X l =1 1 l − ψ  2 S 1 + 1 2  + ψ  1 2  # − 1 2 β  K ∗ ln B ¯ B  (0) . (64) 4.3 S 1 > S 2 F or this case, the auxiliary functions as we ll as the set of non-linear in tegr a l equations ar e exactly the same as in the previous case S 2 = S 1 . The only difference consists in the w ay how the largest eigen v alue Λ ( S 2 ) (0) is expresse d in terms of the auxiliary functions. According to the definition of the Y -function, w e hav e an equation similar to (43) whic h relates Λ ( S 2 ) ( x ) and Y ( S 2 ) ( x ). T his relation can b e written explicitly as Λ ( S 2 ) ( x + i 2 )Λ ( S 2 ) ( x − i 2 ) = f S 2 ( x ) Y ( S 2 ) ( x ) . (65) 23 Applying (44) to (65,53), we obtain ˆ Λ ( S 2 ) ( k ) = i k β 2 cosh [ k / 2] ˆ γ ( k ) + ˆ K ( k ) ˆ Y ( S 2 ) ( k ) , (66) ˆ γ ( k ) = 2 S 1 X l =1 l> ( S 1 − S 2 )+ a e −| k | ( S 2 − S 1 + l − 1 2 ) − 2 S 1 X l =1 l< ( S 1 − S 2 ) − a e −| k | ( S 1 − S 2 − l + 1 2 ) − e −| k | ( 1 2 + a ) , (67) where a = 0 when S 1 − S 2 is an integer n umber and a = 1 / 2 when S 1 − S 2 is a half- in teger n um b er. Here, we recall that the p ossibilit y l = S 1 − S 2 w as already excluded in the definition of the L -o p erator (2 0 ). After p erforming the in v erse F ourier transform and in tegr a tion o v er x , we obtain ln Λ ( S 2 ) ( x ) = β ǫ ( S 1 ,S 2 ) ( x ) +  K ∗ ln Y ( S 2 )  ( x ) , (68) with ǫ ( S 1 ,S 2 ) ( x ) = R ∞ −∞ ˆ γ ( k ) e i kx dk 2 cosh [ k / 2] . A t the part icular p oin t x = 0 , ǫ ( S 1 ,S 2 ) ( x ) is giv en by ǫ ( S 1 ,S 2 ) (0) = ψ  S 1 + S 2 + 1 2  − ψ  S 1 − S 2 + 1 2  . (69) Lastly , the contribution to the free energy is written in terms of the auxiliary function f ( S 2 ,S 1 ) = 1 2 " 2 S 1 X l =1 ∗ 1 S 2 − S 1 + l − ψ  S 1 + S 2 + 1 2  + ψ  S 1 − S 2 + 1 2  # − 1 2 β  K ∗ ln Y ( S 2 )  (0) . (70) It is in teresting to compare ǫ ( S 1 ,S 2 ) ( x ) ( 69) with the previous cases (5 7 ,63). These express ions can b e naturally written in a unified form as follows ε ( S 1 ,S 2 ) = ǫ ( S 1 ,S 2 ) (0) = ǫ ( S 2 ,S 1 ) (0) = ψ  S 1 + S 2 + 1 2  − ψ  | S 1 − S 2 | + 1 2  . (71) 24 According to (1 8), the fr ee energy o f alternating spin c hains is describ ed by the sum of ln Λ ( S 2 ) (0) and ln Λ ( S 1 ) (0). As a result of that, the sum of ε ( S 1 ,S 2 ) and ε ( S 1 ,S 1 ) is the ground state energy of the quan tum Hamiltonia n H ( S 1 ,S 2 ) , ǫ 0 = ψ  S 1 + S 2 + 1 2  − ψ  | S 1 − S 2 | + 1 2  + ψ  2 S 1 + 1 2  − ψ  1 2  , (72) whic h is in a greemen t with the results based on the 2 S -string hypothesis for the cases S 1 = 1 / 2 , S 2 = S [15] and S 2 = S 1 = S [23]. The tota l free energy is the sum of t wo pieces f = f ( S 2 ,S 1 ) + f ( S 1 ,S 1 ) . As w e hav e seen, the term f ( S 2 ,S 1 ) at finite temp erature can b e written a s f ( S 2 ,S 1 ) = f ( S 2 ,S 1 ) 0 − 1 2 β               K ∗ ln B ¯ B  (0) , if S 1 < S 2  K ∗ ln B ¯ B  (0) , if S 1 = S 2  K ∗ ln Y ( S 2 ,S 1 )  (0) , if S 1 > S 2 , (73) where f ( S 2 ,S 1 ) 0 = 1 2 h P ∗ 2 S 1 l =1 1 S 2 − S 1 + l − ε ( S 2 ,S 1 ) i . Here w e ha ve to remind that all auxiliary functions, including B ( x ) and ¯ B ( x ), are differen t fo r differen t cases S 1 < S 2 and S 1 ≥ S 2 . W e lik e to men tion that results of an analysis similar to that ab ov e we re published in [26] for t he study of single Kondo impurities. In the presen t study o f bulk pro p erties of lattice mo dels, t he in tegral equations share some algebraic structures with those in [26], but ha ve rather different analytic prop erties with resp ect to t he driving terms. 5 Numerical res ults In this section, we presen t the n umerical r esults obtained for t he sp ecific heat and mag netic susceptibilit y for the cases S 1 < S 2 , S 1 = S 2 and S 1 > S 2 . 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 c(T) T (S,S) case S=1/2 S=1 S=3/2 S=2 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 χ (T) T S=1/2 S=1 S=3/2 S=2 Figure 1 : Sp ecific heat c ( T ) and χ ( T ) magnetic susceptibilit y v ersus temp er- ature T for S = 1 / 2 , 1 , 3 / 2 , 2. W e ha ve solv ed num erically the non- linear in tegral equations by iterat io n. The conv olutions ha ve b een calculated in F ourier space using the F ast F ourier T ransform algorithm (FFT). Ev entually , w e hav e obtained the free energy as a f unction of temp erature and magnetic field. Instead of p erforming num erical differen tiat io ns to obtain the deriv ativ es of the free energy with resp ect to temp erature and magnetic field, w e hav e used asso ciated integral equations for the deriv ativ es of the auxiliary f unc- tions. These integral equations ar ise from the differen tiation of the set of 26 non-linear equations, e.g. with resp ect to the temp erature T . 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 2 4 6 8 10 c(T) T (1/2,S) case S=1/2 S=1 S=3/2 S=2 0 2 4 6 8 10 0 2 4 6 8 10 χ (T) T S=1/2 S=1 S=3/2 S=2 Figure 2 : Sp ecific heat c ( T ) and χ ( T ) magnetic susceptibilit y v ersus temp er- ature T for S 1 = 1 / 2 and S 2 = S = 1 / 2 , 1 , 3 / 2 , 2. Lastly , we ha ve used the relation among the deriv ativ es of the auxiliary functions reading ∂ ∂ T ln B ( x ) = b ( x ) 1 + b ( x ) ∂ ∂ T ln b ( x ) , (74) ∂ 2 ∂ T 2 ln B ( x ) = b ( x ) 1 + b ( x ) " 1 1 + b ( x )  ∂ ∂ T ln b ( x )  2 + ∂ 2 ∂ T 2 ln b ( x ) # . (75) This w ay , we obtained for eac h incremen t in the order o f differen tiation a 27 new set of linear in tegral equations, where the low er order deriv ativ es app ear just as co efficien ts. In Fig ur es 1-3, w e sho w the sp ecific heat and t he magnetic susceptibilit y as functions of temp erature for the particular cases S 1 = S 2 = S , S 1 = 1 / 2 , S 2 = S and S 1 = S, S 2 = 1 / 2 for S = 1 / 2 , 1 , 3 / 2 , 2, resp ectiv ely . The system sho ws a n tiferro ma g netic b eha viour fo r the first case S 1 = S 2 . A t lo w temp erature c ( T ) presen ts a linear t emp erature dependence and χ ( T ) approac hes a finite v alue. F o r the case S 2 > S 1 , w e ha v e finite magnetization M f = S 2 − S 1 2 at zero temp erature and v anishing magnetic field ( T = 0 , h = 0 + ) in agreemen t with [1 8]. In the other limit ( T = 0 + , h = 0) , we hav e zero magnetization. This is compatible with the fact that at low temp erature and zero ma g netic field χ ( T ) shows div ergen t b eha viour. F or finite (ev en small) magnetic field the system b ecomes p olarized presen ting finite magnetization asso ciated with a drop of χ ( T ). In the last case, S 1 > S 2 , the system b eha v es as an an tiferromagnet. It ha s zero magnetization in b oth limits ( T = 0 , h = 0 + ) a nd ( T = 0 + , h = 0) in accordance with [17]. F or the cases S 2 > S 1 and S 2 < S 1 , the mo dels presen t residual en tropy . The sp ecific v alues fo r t his quan tity can b e extracted from low temp erature asymptotic solutions of the non-linear integral equations. The results are giv en b y S r es = 1 2 ln [2( S 2 − S 1 ) + 1] and S r es = 1 2 ln  sin π (2 S 2 +1) 2 S 1 +2 sin π 2 S 1 +2  for S 2 > S 1 and S 2 < S 1 resp ectiv ely . The latter case was considered in [17] for ( S 1 = 1 , S 2 = 1 / 2) using the TBA approac h. There, how ev er, the exact v alue of the residual en tropy was left op en due to limitations of their metho d. 28 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 2 4 6 8 10 c(T) T (S,1/2) case S=1/2 S=1 S=3/2 S=2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 8 10 χ (T) T S=1/2 S=1 S=3/2 S=2 Figure 3 : Sp ecific heat c ( T ) and χ ( T ) magnetic susceptibilit y v ersus temp er- ature T for S 1 = S = 1 / 2 , 1 , 3 / 2 , 2 and S 2 = 1 / 2 . 6 Thermal cu rren t In this section, w e a r e interes t ed in the thermal Drude w eight D th ( T ) at finite temp erature. W e restrict ourselv es t o the case S 1 = S 2 , where the thermal curren t is related to the second conserv ed c harge (5 ) o f t he t r a nsfer matrix. Sp ecifically , w e consider the lo cal conserv ation of energy in terms of a con tinuit y equation. This relates the time deriv ativ e of the lo cal Hamiltonian H ii +1 to the dive rgence of the thermal curren t j E , ˙ H = −∇ j E . Here, the 29 lo cal term H ii +1 stands for H ii +1 = P i,i +1 ∂ ∂ λ L ( S 1 ,S 1 ) i,i +1 ( λ )    λ =0 , H = L X i =1 H ii +1 . (76) As the time deriv ativ e leads t o the comm utator with the Hamiltonian, w e obtain ˙ H i,i +1 = i [ H , H i,i +1 ( t )] = − i  j E i +1 ( t ) − j E i ( t )  , (7 7 ) where the lo cal energy curren t j E i is giv en b y j E i = i [ H i − 1 i , H ii +1 ] , (78) and the to tal thermal curren t is J E = P L i =1 j E i . On t he other hand, just by comparing the expression for J E and the second logarithmic deriv ativ e of the transfer matrix J (2) , w e obtain J E = J (2) + i L 2 ∂ 2 ∂ λ 2 ζ S 1 ,S 1 ( λ )    λ =0 . (79) The transp ort co efficien ts are determined fr o m the Kub o for m ula [2 7] in terms of the exp ectation v alue of the thermal curren t J E , suc h that [28, 2 9] D th ( T ) = β 2  J 2 E  . (80) In order to calculate the exp ectation v alue hJ 2 E i , we in tro duce a new partition function ¯ Z as, ¯ Z = T r  exp  − β H − λ n J ( n )  . (81) In this w ay , w e obtain the exp ectation v alues of J (2) through the logarithmic deriv ativ e of ¯ Z ,  ∂ ∂ λ 2  2 ln ¯ Z    λ 2 =0 =  J 2 E  , (82) 30 where w e used the fact that the exp ectation v alue o f the thermal curren t in thermo dynamical equilibrium is zero hJ E i = 0. T o compute the partition function ¯ Z , we consider the pro cedure dev elop ed in [29]. W e rewrite the partition function ¯ Z in terms of the row-to-ro w transfer matrix suc h that ¯ Z = lim N →∞ T r  exp  T ( u 1 ) . . . T ( u N ) T (0) − N  , = T r " exp lim N →∞ N X l =1 { ln T ( u l ) − ln T (0) } !# . (83) The num b ers u 1 , . . . , u N are chosen in suc h a wa y that the following relation is satisfied, lim N →∞ N X l =1 { ln T ( u l ) − ln T (0) } = − β ∂ ∂ x ln T ( x )    x =0 + λ n i n − 1 ∂ n ∂ x n ln T ( x )    x =0 . (84) W e can pro ceed analog o usly to section 2 and in tro duce a quan tum transfer matrix asso ciated t o the partition function ¯ Z . Instead of the staggered v ertex mo del with alternation in v ertical direction b et wee n T ( − τ ) and T ( − τ ), w e ha ve now N differen t terms of the fo rm T (0) − 1 T ( u l ) for l = 1 , . . . , N . As T (0) − 1 = T (0) / N , we can write T (0) − 1 = [(2 S 1 )!] − 2 L T ( − ρ ). So, w e hav e the alternation of T ( − ρ ) and T ( u l ) whic h is a sp ecial case of the previous sections. Therefore, w e can pro ceed along the same lines as b efore which is equiv- alen t to substitute φ + ( x ) → Q N l =1 φ l ( x ) and φ − ( x ) → Q N l =1 φ 0 ( x ) where φ l ( x ) = x − i u l and φ 0 ( x ) = x . In this wa y , the partition function can b e written in the thermo dynamical 31 limit in terms of the largest eigen v alue, lim L →∞ 1 L ln ¯ Z = ln Λ(0 ) , (85) whic h is written as ln Λ(0) = ( − β + λ n ∂ n − 1 ∂ x n − 1 ) E ( x )    x =0 +  K ∗ ln B ¯ B  (0) , (86) where E ( x ) = ǫ ( S 1 ,S 1 ) ( x ). The auxiliary functions B and ¯ B satisfy the following set of non- linear in tegral equations            ln y ( 1 2 ) ( x ) . . . ln y ( S 1 − 1 2 ) ( x ) ln b ( x ) ln ¯ b ( x )            =            0 . . . 0 ( − β + λ n ∂ n − 1 ∂ x n − 1 ) d ( x ) ( − β + λ n ∂ n − 1 ∂ x n − 1 ) d ( x )            + K ∗            ln Y ( 1 2 ) ( x ) . . . ln Y ( S 1 − 1 2 ) ( x ) ln B ( x ) ln ¯ B ( x )            . (87) Therefore, the thermal Drude we igh t is giv en by , D th ( T ) = β 2 D J (2) 2 E = β 2  ∂ ∂ λ 2  2 ln Λ(0)    λ 2 =0 . (88) In Fig ure 4 , w e show the thermal Drude w eight as function of the tem- p erature for S 1 = S 2 = S . It exhibits a linear b eha viour at lo w temp eratures and is prop ortional to the cen tral charge c = 3 S S +1 . This is in agr eemen t with the spin-1 / 2 case [29]. Before closing this section, w e w o uld lik e to mention that in the general case ( S 1 , S 2 ) the thermal curren t do es not lo ok lik e a conserv ed current. In this case, w e cannot pro vide exact results for t he D rude weigh t. Nev erthe- less, w e are able to pro vide an exact description of the second logarithmic 32 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 D th (T) T S=1/2 S=1 S=3/2 S=2 Figure 4: Thermal Drude w eight D th ( T ) as function of temp erature for S = 1 / 2 , 1 , 3 / 2 , 2 . deriv ativ e of the transfer matrix. How ev er, the phys ical in terpretation o f this quan tity has eluded us so far. 7 Conclus ion In this pap er we mana ged to construct the quan tum transfer matr ix for the case of non-isomorphic a uxiliary and quan tum spaces of in t eracting spins . W e considered explicitly the generic ( S 1 , S 2 ) case of alternating spin c hains 33 and obta ined a finite set of non-linear inte gral equations. Thes e equations w ere solv ed numerically for the cases S 1 < S 2 and S 1 ≥ S 2 . In this w a y , w e obtained the sp ecific heat and the magnetic susceptibilit y as functions of temp erature. F o r the par ticular case S 1 = S 2 , w e also pro vided results for the thermal Drude w eight at finite temp erature. The system b eha v es antiferromagnetically for S 1 ≥ S 2 and presen ts finite magnetization in the remaining case S 1 < S 2 . In terestingly , for all S 1 6 = S 2 w e hav e residual entrop y at zero temp erature whic h w e w ere able to ev aluate exactly . 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