Ground state and low excitations of an integrable chain with alternating spins St Meinerz and B - D Dorfelx Institut fur Physik, Humboldt-Universitat, Theorie der Elementarteilchen Invalidenstrae 110, 10099 Berlin, Germany Abstract An anisotropic integrable spin chain, consisting of spins s = 1 and s = 1, is investigated [1] It is characterized by two real parameters c and ~c, the coupling constants of the spin interactions For the case c < 0 and ~c < 0 the ground state conguration is obtained by means of thermodynamic Bethe ansatz Furthermore the low excitations are calculated It turns out, that apart from free magnon states being the holes in the ground state rapidity distribution, there exist bound states given by special string solutions of Bethe ansatz equations (BAE) in analogy to [13] The dispersion law of these excitations is calculated numerically PACS numbers: 7510 JM, 7540 Fa Short title: Z( 1 ; 1) Heisenberg spin chain September 14, 1995 z email: meissner@qftphysikhu-berlinde x email: doerfel@ifhde
1 Introduction Since the development of the quantum inverse scattering method (QISM) [, 3] many generalisations of the well known Z( 1 ) model have been investigated [4, 5, 6] The appearance of interesting new features of these models made those analysises very fruitful In 199 de Vega and Woynarovich suggested a new kind of generalisation by constructing a model containing spin- 1 - and spin-1-particles [1] In comparison with models containing only one kind of spin it shows a richer structure due to additional parameters, the spin interaction coupling constants, making an investigation worthwhile In this paper we want to study the Z( 1 ; 1) model with strictly alternating spins Denitions are reviewed in section In section 3 we carry out the thermodynamic Bethe ansatz (TBA) and obtain the ground state for negative coupling of the spin interactions Section 4 deals with the higher level Bethe ansatz for low excitations above this ground state and section 5 contains our conclusions Description of the model We consider the Hamiltonian of a spin chain of length N [1] H() = c H() + ~c ~ H()? HS z ; (1) where H couples two spins s = 1 and one spin s = 1, while the converse is true for ~H (for explicit construction see again [1], bars and tildes are interchanged with respect to this reference as in subsequent publications of de Vega et al) H is an external magnetic eld We impose periodic boundary conditions The Hamiltonian contains an Z type anisotropy, which is parametrized by e i or e? respectively We investigate the weak antiferromagnetic case, ie the parametrization is e i and we restrict ourselves to 0 < < = Moreover we have two additional real parameters ~c and c being the coupling constants of the spin interactions and dominating the qualitative behaviour of the model The Bethe ansatz equations (BAE) determining the solution of the model are sinh( j + i ) sinh( j? i ) sinh( j + i) sinh( j? i)! N =? MY k=1 sinh( j? k + i) ; j = 1 : : : M: () sinh( j? k? i) One can express energy, momentum and spin in terms of BAE roots j : E = c E + ~c E ~ 3N?? M H; M E =? j=1 sin cosh j? cos ;
~E =? P = i M j=1 M j=1 sin cosh j? cos ; (3) ( log sinh( j + i )! sinh( j? i ) + log sinh( j + i) sinh( j? i)!) 3 ; (4) S z = 3N? M: (5) Here we have substracted a constant in the momentum in order to make this magnitude vanishing for the ferromagnetic state Furthermore additive constants in H and ~ H are dropped due to dierent normalisation compared with [1] 3 Thermodynamic Bethe ansatz (TBA) and ground state for negative coupling constants We want to study the model in the thermodynamic limit N! 1 Then only solutions of the string type occur (for M x): Here n; n;j; = n; + i(n + 1? j) + 1 i(1? ) + n;j; ; j = 1 : : : n: (31) 4 is the real center of the string, n is the string length and the parity of the string with values 1 The last term is a correction due to nite size eects Now the permitted values of the string length n and related parities depending on the anisotropy are to determine Several approaches have been used to obtain them for the massive Thirring model [8] and for the Z( 1 ) [7] and Z(S) models [10, 11, 1] all leading to the Takahashi-Korepin conditions, a system of inequalities describing admissible string length and parities: n sin j sin (n? j) > 0; j = 1 : : : n? 1: (3) In the Z(S) case additional restrictions on the spin arise By applying similar methods to our model one can show that fullling the Takahashi- Korepin conditions is necessary and sucient for an admissible pair (n; n ) for the number of magnons M x On the other hand we did not succeed in deriving such conditions in the case N; M! 1; M=N x from the BAE directly Nevertheless there is a series of arguments favouring the conjecture that in this case the same strings as in the previous one can exist In the following we want to state a few of them briey Generally the string hypothesis is believed to be valid in the case N; M! 1; M=N x (ie string corrections are exponentially small) for non-vanishing external magnetic eld (see eg [9]) For zero magnetic eld the numerical results of Frahm et al show that string corrections are of orders O(1=N) to O(1), what does not aect the thermodynamics of the model
4 Furthermore Takahashi and Suzuki obtained a phenomenological rule determining admissible strings for the Z( 1 ) model proven to be equivalent to the Takahashi- Korepin conditions Fullling this criterion requires that a singlet state can be constructed by introducing a maximum number of strings of one type Kirillov and Reshetikhin generalized this criterion naturally to higher spins In the Z(S) it is as well a selection rule for permitted spin values as a criterion for an admissible pair (n; n ) In the following we therefore shall assume the existence of strings fullling the Takahashi-Korepin conditions Substituting (31) into () and taking the logarithm yields with the known notations Nt j;1 ( n j ) + Nt j;( n j ) = In j + k t j;s () = min(n j ;S) k=1 jk () = f(; jn j? n k j; j k ) and jk ( n j? n k ; j k ) (33) f(; jn j? Sj + k? 1; j ); (34) min(n j ;n k )?1 +f(; (n j + n k ); j k ) + f(; jn j? n k j + k; j k ); (35) f(; n; ) = ( k=1 0 n= Z arctan((cot(n=)) tanh ) n= = Z : (36) Here we have used that a given string length n > 1 corresponds to a unique parity, what is a consequency of (3) The numbers I n j are half-odd-integers counting the strings of length n j Introducing particle and hole densities in the usual way (see eg [5] eqs 60-63) we perform the limiting process N! 1 a j;1 () + a j; () = ( j () + ~ j ())(?1) r(j) + k T jk k (); (37) where a b() denotes the convolution and a b() = Z 1?1 a j;s () = 1 da(? )b() (38) d d t j;s(); T j;k () = 1 d d j;k(): (39) The sign (?1) r(j) results from the requirement of positive densities in the 'noninteracting' limit (ie only one string type is present)[1]
The analysis of (3) remains complicated for arbitrary [7], but at special values the picture becomes easy, namely = =; : : : integer, 3 Following Babujian and Tsvelick [6] we want to consider that case only Due to periodicity we are left with strings (i) n j = j; j = 1 : : :? 1; j = 1, (ii) n = 1; =?1 Equation (37) then reduces to a j;1 () + a j; () = j () + ~ j () + k=1 a ;1 () + a ; () =? ()? ~ () + T jk k (); j = 1 : : :? 1; k=1 T k k (): (310) We are now able to express energy, momentum and spin in terms of the densities via (3), (4) and (5) The standard procedure leads to equations determining the equilibrium state at temperature T : T ln with 1 + exp j T = Hn j? ca j;1 ()? ~ca j; () + k=1 T ln 1 + exp?k T 5 A jk ()(311) A jk () = (?1) r(j) T jk (; j k ) + () jk (31) and ~ j j = exp j T : (313) Again the free energy can be expressed in terms of our new variables j (): F = F Z 1 N =??j d (?1) r(j) (a j;1 () + a j; ())T ln 1 + exp? 3H?1 T : (314) j=1 Reversing the operator A jk in (311) by applying C jk = () jk? s()( j+1k + j?1k ); j; k = 1 : : : (315) with s() = 1 cosh(=) (316)
6 yields with 1 () () = T s ln(f( ))? cs(); = T s ln(f( 3 )f( 1 ))? ~cs(); j () = T s ln(f( j+1 )f( j?1 )) + j? T s ln(f(? )); j = 3 : : :? ;?1 () = H + T s ln(f(?)); () = H? T s ln(f(?)); (317) f(x) = 1 + e x=t : (318) From these equations it can be easily established that j > 0 for j = 3 : : :? 1 Since we are interested in the ground state we take the limit T! 0 in equations (311) and (314) Taking into account that j > 0 for j = 3 : : :? 1 we have + j () =?ca j;1 ()? ~ca j; ()?? 1 A j1 ()?? A j ()?? A j (); (319) F = F N = Z 1?1 d h (a 1;1 () + a 1; ())? 1 () +(a ;1 () + a ; ())? ()? (a ;1() + a ; ())? ()i? 3H ; (30) where + j and? j denote positive and negative parts of the function j respectively Now we want to discuss equation (319) for the case H = 0 + in dependence of the signs of c and ~c (i) c > 0; ~c > 0 This sector has been investigated completely by de Vega and Woynarovich [1] The solution is (ii) c > 0; ~c < 0 j () =?cs() j1? ~cs() j : (31) The picture in this sector is dicult, since the quadrant is divided by a phase line starting in the origin and going to innity, which we obtained from the TBA equations In the lower area the model behaves as in case (iv), while in the upper one we expect for the ground state a mixture of strings of length 1 with dierent parities and nite Fermi zones We hope to return to this case in greater detail later
(iii) c < 0; ~c > 0 The situation in this sector is similar to the case above Again we have a phase line producing one area of case (iv) type and another one where the ground state is expected to be formed by the (; +)- and (1;?)-strings having again nite Fermi zones (iv) c < 0; ~c < 0 In this case 1 () and () are positive The only string present in the ground state is (1;?) Equations (319) and (30) reduce to + j () =?ca j;1 ()? ~ca j; ()?? A j (); (3) F = F N =? Z 1?1 d(a ;1 () + a ; ())? (): (33) The solutions of (3) can be given via their Fourier transforms: cosh[p(? )=] ^ 1 (p) =?c cosh[p(? 1)=] cosh[p(? j? 1)=] ^ j (p) =?c cosh[p(? 1)=] 1 ^?1 (p) =?c? 3)=]? ~ccosh[p( cosh[p(? 1)=] ;? ~c cosh(p=) cosh[p(? j? 1)=] ; cosh[p(? 1)=] cosh[p(? 1)=]? ~c cosh(p=) cosh[p(? 1)=] ; 1 ^ (p) = c cosh[p(? 1)=] + ~c cosh(p=) cosh[p(? 1)=] : (34) The ground state energy is F = F Z 1 N = c dp sinh(p=) + sinh(p)?1 cosh[p(? 1)=] sinh(p=) Z 1 [sinh(p=) + sinh(p)] cosh(p=) +~c dp : (35)?1 cosh[p(? 1)=] sinh(p=) For c = ~c the continuum limit provides a conformal invariant theory Due to the existence of one type of elementary excitations the central charge of the Virasoro algebra is c = 1 In the following we shall consider the case (iv) only 7 4 Higher level Bethe ansatz for the low excitations In this section we want to derive equations for excitations above the ground state in the case c < 0; ~c < 0 Starting point of our analysis is the result of section 3 Though
8 we have derived the ground state conguration of BAE roots only for the special case = =; integer, we extend this result to the whole range of ; 0 < < =, motivated by the results of Frahm et al [1], who found for the Z(S) Hamiltonians, where only special intervals of the anisotropy are permitted, that the ground state conguration does not change within any of these intervals Since in our case no restrictions on arise, ie we have only one permitted interval, we expect the ground state conguration to be the one obtained for the above mentioned special values in the whole -range We write down the BAE for this ground state in the limit N! 1 with a nite number of excitations (holes in the ground state rapidity distribution and additional complex roots): 0 (; =) + 0 (; ) =?()? 1 N N h h=1 (? h ) Z 1 +?1 d0 ( 0 ) 0 (? 0 ; ) + 1 N Here we have introduced the new notations ; n = 1 f(; n; +1); N l l=1 0 (? z l ; ): (41) ; n = 1 f(; n;?1) (4) and the prime means the derivative with respect to the rst argument The numbers of holes h and additional complex roots z l are denoted by N h and N l respectively The hole positions are dened as solutions of (33) for the omitted I h The solution of (41) contains dierent contributions obtained by Fourier transformation: with () = 0 () + 1 N ( h() + c () + w ()) (43) ^ 0 (p) = ^ h (p) =? ^ c (p) =? ^ w (p) = 1 + cosh(p=) cosh(p(? )=) N h h=1 N w= l=1 N c= l=1 e?ip h sinh(p=) sinh(p=) cosh(p(? )=) e?ip l cosh(p=) cosh(p l) cosh(p(? )=) e?ip sinh(p(? )=) cosh(p(? l)=) l : (44) cosh(p(? )=) sinh(p=) Here we have distinguished between complex close (jim(z l )j < =? ) and wide roots (jim(z l )j > =? ) diering in their Fourier transforms Moreover we have used that complex roots appear in conjugated pairs (z l ; z l ) only
9 Energy and momentum of the elementary hole excitations are given through p h ( h ) = arctan " h ( h ) = " h ( h ) + ~" h ( h ); " h ( h ) =? c 1? cosh( h =(? )) ; ~" h ( h ) =? 4~c cos(=(? )) cosh( h =(? ))? cosh( h =(? )) + cos(=(? )) ; (45) sinh h?!! + + arctan sinh( h=(? )) cos(=(? ))! + : (46) We did not nd an explicit expression for the dispersion law of hole excitations Nevertheless, numerical calculation suggests, that the curve shows the expected behaviour (cf gure ) Now we want to derive equations for hole positions and complex roots excluding the vacuum parameters by using the method of Babelon et al [13] For this purpose we write down the BAE in integral approximation for a complex root z: exp(i[i(z)? F (z)]) =?1 (47) with I 0 (z) = N Z 1?1 d 0 ( 0 ) 0 (z? 0 ; ); (48) F 0 (z) = N 0 (z; 1 ) + N0 (z; ) + N h h=1 0 (z? h ; ) + N c l=1 0 (z? z l ; );(49) and () is the regular density () = () + 1 N N h h=1 (? h ): (410) I 0 (z) has discontinuities on the lines Im(z) = (=? ) So analysing (47) we have to consider three cases a) Im(z l ) > =? We can evaluate the function I(z)? F (z) directly by integrating Noticing I 0 (z)? F 0 (z) =? z? i N: (411) I(1)? F (1) = 0 (41) one gets I(z)? F (z) =?N 1 u? i du (413)
10 and exp?in u? i du =?1: (414) 1 Splitting (u) into vacuum and excitation contributions u? i = 0 u? i we get for (414) exp + 1 N u? i (415) in 0 u? i du + i u? i du =?1: (416) 1 1 The rst exponent term contains an explicit N-dependence, while the second does not So the validity of the above equation in the limit N! 1 requires, that the second term cancels the N-dependence in the real part of the rst one by approaching z in exponential order towards a pole of the integrand These poles of I 0 (u)? F 0 (u) are with the relevant ones z l i; h ( i? i); 1 i; i; (417) z l i with residue i: (418) Now it can be easily established that Im 0 u? i du 1 > 0: (419) Therefore it must exist z l with z = z l + i b)?=+ < Im(z l ) < =? Here evaluating (48) by deforming the integration contour provides an additional term due to residue theorem One has to replace I(z)! I(z)? N z? i + i : (40) Then (47) reads exp in u? i? z? i 1 + i du =?1: (41) Using the Fourier transforms (44) one can establish the relation 0 (u) + 0 (u? i + i) = 0: (4) Then determining the sign of the real part of the exponent leads to Im 0 u + i + 0 u? i ( > 0 : Im(z) > 0 du 1 < 0 : Im(z) < 0 (43) and we conclude that there exists z l with z = z l +i for Im(z) > 0 and z l with z = z l?i for Im(z) < 0 respectively
c) Im(z l ) <?= + Evaluating (48) again by contour deformation and substituting into (47) leads to exp in u? i? z? i 1 + i + z + i? i du =?1: (44) Using (4) the N-dependent exponent term reduces to in 0 u + i du: (45) 1 With Im 0 u + i du < 0 (46) 1 follows z = z l? i, where z l is a BAE root Now we are able to determine the congurations in which complex roots can appear We distinguish three cases for the anisotropy: a) 0 < < =4 Because of symmetry with respect to the real axis we only consider Im(z) > 0 This leads to the existence of z l with z = z l + i Repeating this argument for z l (Im(z l ) > 0 because of < =4) generates another root z l? i This process generating roots breaks down if the last generated root has negative imaginary part Therefore the 'minimal' conguration induced by z is a chain with spacing i situated completely in the upper half plane except the last generated root For further discussion one has to distinguish wide and close roots 11 (i) Im(z) > =? Taking the product over equations (414) and (41) respectively for all chain members shows that there remains a non-compensated N-dependent term until the chain is continued into the lower region of wide roots Then the conguration gets 'stable' in the limit N! 1 Notice that also the conguration with all roots complex conjugated exists, so that the conguration is actually a double chain (ii) 0 < Im(z) < =? Here the above procedure does not lead to a 'stable' conguration The only way for this conguration to exist in the limit N! 1 is arranging the roots symmetrically with respect to the real axis Therefore this conguration is of the string type It coincides with its complex conjugated one (see gure 1) b) =4 < < =3 and c) =3 < < = Analogous argumentations lead again to short double chains and strings (see gure 1) We can write down the possible congurations for the three cases in a more compact way: (i) z t = z? it; t = 0 : : : n; n = h i ; h i? 1, z is a wide root
1 = 0?= Figure 1 Some special congurations of complex roots for the three cases of With '' a member of the 'minimal' conguration is symbolized and '' denotes additional roots to make the conguration 'stable' String conguration members are denoted by '' too due to the fact that stability is realized by symmetrical arrangement The lines varying in the three cases denotes (=? ) Notice that for non-symmetric congurations the complex conjugated one in this picture is omitted h i (ii) z t = z? it; t = 0 : : : n; n n max ; n max =? ; Im(z) = n, z is a close root The higher level BAE for a state with complex root congurations of the rst type can be obtained by taking the product over equations (47) for all members and calculating I(z)? F (z) directly via Fourier transforms With the new parameters l = z l? in l ; for n even; ~ l = z l? in l ; for n odd; P : : : number of -congurations; ~P : : : number of ~-congurations; =? (47) it follows YN h h=1 sinh(( l? h ) + i=) PY sinh(( l? h )? i=) =? sinh(( l? j ) + i) sinh(( l? j ) + i) j=1 ~PY j=1 cosh(( l? ~ j ) + i) cosh(( l? ~ j ) + i) ;
YN h h=1 cosh((~ l? h ) + i=) PY cosh((~ l? h )? i=) =? cosh((~ l? j ) + i) cosh((~ l? j ) + i) j=1 ~PY j=1 13 sinh((~ l? ~ j ) + i) sinh((~ l? ~ j ) + i) : (48) Calculating energy and momentum for the congurations results in vanishing of these magnitudes for the rst type and E =?c 4 sin [(n + 1)=(? )] cosh [=(? )]? cosh [=(? )]? cos [(n + 1)=(? )] (?~c 4 sin [(n + )=(? )] cosh [=(? )]? cosh [=(? )]? cos [(n + )=(? )] P = i=0 sin [n=(? )] cosh [=(? )] + (49) cosh [=(? )]? cos [n=(? )]! sinh [=(? )] arctan + 3 (430) sin [(n + i)=(? )] for the second one (n 1) The dispersion law of these excitations has been calculated numerically for dierent < =3 and string length n and c=~c = 1 A comparison with the dispersion law of a free two magnon state, where the momenta of the magnons are equal as given in gure reveals, that these excitations can be identied with bound magnon states in analogy to [13] One can see from the picture, that the maximum of the curve decreases monotonically with increasing n up to n max for x, approaching the dispersion curve for the free state On the other hand for! =3, where only the n = 1 state exists, again the bound state curve approaches the one for the free state monotonically Moreover there is a state with n = 0 allowed through the whole -range Its energy and momentum are given via (49) and (430), where in the last formula the term for i = 0 is dropped This leads to a range for the momentum between those for a single hole excitation and a bound state Therefore the bound state interpretation fails ) 5 Conclusions We have investigated a generalized Heisenberg spin chain with alternating spins Z( 1 ; 1) in the gapless region By means of thermodynamic Bethe ansatz (TBA) integral equations determining the ground state have been derived In the case of negative coupling of spin interactions these equations were solved It turns out that the ground state is formed by a sea of (1;?)- strings and is therefore of antiferromagnetic nature Weakly excited states above this antiferromagnetic vacuum have been analysed following the method introduced in [13] In analogy to the Z ( 1 ) model for?1 < < 0 two types of excitations appear The rst one are scattering states
14 4 4 E=jcj 16 8 0 0 4 8 1 16 0 P; = 0:9; n = 1 4 16 8 0 0 4 8 1 16 0 P; = 0:5; n = 1 4 E=jcj 16 8 0 0 4 8 1 16 0 P; = 0:1; n = 0 16 8 0 0 4 8 1 16 0 P; = 0:1; n = 5 Figure Dispersion relation for Type II excitations (solid line) compared with the dispersion law for a free two magnon state (dotted line) for dierent of magnons Higher level BAE are derived which determine the parameters of these excitations The second type can be identied with bound magnon states in analogy to [13] Acknowledgment We would like to thank H M Babujian for helpful discussions References [1] de Vega H J and Woynarovich F 199 J Phys A: Math Gen 4499 [] Faddeev L D and Takhtajan L A 1979 Usp Math Nauk 34 5 [Russ Math Surveys 34 (1979) 11] [3] Kulish P P and Sklyanin E K 1981 Lecture Notes in Physics vol 151 (Berlin: Springer) 61 [4] Takhtajan L A 198 Phys Lett 87A 479
[5] Babujian H M 1983 Nucl Phys B15 [FS7] 317 [6] Babujian H M and Tsvelick A M 1986 Nucl Phys B65 [FS15] 4 [7] Takahashi M and Suzuki M 197 Progr Theor Phys 48 187 [8] Korepin V E 1979 Theor Math Phys 41 953 [9] Tsvelick A M and Wiegmann P B 1983 Adv in Phys 3 453 [10] Kirillov A N and Reshetikhin N Yu 1985 Zap Nauch Semin LOMI 145 109 [J Sov Math 35 (1985) 109]; Kirillov A N and Reshetikhin N Yu 1985 Zap Nauch Semin LOMI 146 31 [J Sov Math 40 (1988) ]; Kirillov A N and Reshetikhin N Yu 1985 Zap Nauch Semin LOMI 146 47 [J Sov Math 40 (1988) 35] [11] Kirillov A N and Reshetikhin N Yu 1987 J Phys A: Math Gen 0 1565; Kirillov A N and Reshetikhin N Yu 1987 J Phys A: Math Gen 0 1587 [1] Frahm H, Yu N C and Fowler M 1990 Nucl Phys B336 (1990) 396 [13] Babelon O, de Vega H J and Viallet C M 1983 Nucl Phys B0 [FS8] 13 15
16 Figure captions Figure 1 Some special congurations of complex roots for the three cases of With '' a member of the 'minimal' conguration is symbolized and '' denotes additional roots to make the conguration 'stable' String conguration members are denoted by '' too due to the fact that stability is realized by symmetrical arrangement The lines varying in the three cases denotes (=? ) Notice that for non-symmetric congurations the complex conjugated one in this picture is omitted Figure Dispersion relation for Type II excitations (solid line) compared with the dispersion law for a free two magnon state (dotted line) for dierent