Analysis of the Bethe ansatz equations of the XXZ model

Nuclear Physics B220 [FS8] (1983) 13-34 (~) North-Holland Publishing Company ANALYSIS OF T H E B E T H E A N S A T Z XXZ MODEL EQUATIONS OF T H E ...

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Nuclear Physics B220 [FS8] (1983) 13-34 (~) North-Holland Publishing Company

ANALYSIS

OF T H E B E T H E A N S A T Z XXZ MODEL

EQUATIONS

OF T H E

O. BABELON, H.J. de VEGA and C.M. VIALLET Laboratoire de Physique Thgorique et Hautes Energies, Paris*, France Received 28 June 1982 (Revised 25 October 1982) We analyse the Bethe ansatz equations of the X X Z model in the antiferromagnetic region, without assuming a priori the existence of strings. Excited states are described by a finite number of parameters. These parameters satisfy a closed system of equations, which we obtain by eliminating the parameters of the vacuum from the original Bethe ansatz equations. Strings are only particular solutions of these equations.

1. Introduction A n e w a p p r o a c h for t h e analysis of the B e t h e ansatz e q u a t i o n s has b e e n r e c e n t l y p r o p o s e d for t h e chiral i n v a r i a n t G r o s s - N e v e u m o d e l [1]. This w o r k allows a b e t t e r u n d e r s t a n d i n g of t h e e x c i t e d states a b o v e a g r o u n d s t a t e of a n t i f e r r o m a g n e t i c n a t u r e . W e a n a l y s e h e r e , with this a p p r o a c h , t h e e x c i t a t i o n s for t h e a n i s o t r o p i c H e i s e n b e r g m o d e l ( X X Z m o d e l ) , in t h e a n t i f e r r o m a g n e t i c case. T h e h a m i l t o n i a n is H =-~

x

X

y

y

z°z

, 10rnO"n+l "[-OrnO'n+l J~'-AO'nO'n+I

"

(1)

w h e r e o-x, cr y, z a r e t h e P a u l i m a t r i c e s , a n d we shall s u p p o s e the n u m b e r of sites N to b e even. A s is w e l l - k n o w n [2], t h e p r o b l e m of d i a g o n a l i z i n g H r e d u c e s b y t h e B e t h e a n s a t z to t h e r e s o l u t i o n of a s y s t e m of c o u p l e d t r a n s c e n d e n t a l e q u a t i o n s . E a c h e i g e n s t a t e is c h a r a c t e r i s e d b y a set of r o o t s of t h e s e e q u a t i o n s . In the t h e r m o d y n a m i c limit (N--> oo), t h e g r o u n d s t a t e is a s s o c i a t e d to a c o n t i n u o u s d i s t r i b u t i o n of real roots. E x c i t e d states c o r r e s p o n d to s o m e d i s t r i b u t i o n of r e a l r o o t s plus a finite n u m b e r of c o m p l e x roots. In this limit, t h e s y s t e m of t r a n s c e n d e n t a l e q u a t i o n s gives (i) an i n t e g r a l e q u a t i o n for the d e n s i t y of real r o o t s a n d (ii) a finite n u m b e r of c o u p l e d e q u a t i o n s for t h e c o m p l e x r o o t s , involving also t h e real roots. It was u s u a l l y a s s u m e d t h a t t h e c o m p l e x r o o t s a p p e a r in strings [2, 3]. In the p r e s e n t a p p r o a c h , no such h y p o t h e s i s is m a d e . W e first solve t h e i n t e g r a l e q u a t i o n * Laboratoire Associ6 au CNRS no. 280. Postal address: Universit6 P. et M. Curie, Tour 16 - ler dtage, 4, place Jussieu, 75230 Paris Cedex 05, France. 13

14

O. Babelon et al. / Bethe ansatz equations

for arbitrary complex roots and then use the solution in the remaining equations to get a closed system for the complex roots. It is remarkable that this system has again the typical form of the Bethe ansatz equations, suggesting the existence of a higher level Bethe ansatz [1]. Our analysis differs from [1] in that we use m o r e directly the integral equations and the analytic properties of its solution, thus having a more straightforward derivation of the final equations. The description of the ground state and the excited states depends on the value of A. For the ground state we must distinguish A < - 1 , - 1 < A < 1, and A > 1 [2]. For A > 1 the system is ferromagnetic and we do not consider this case here. The analysis of the excitations for A < 1 requires a further distinction between - 1 < A < 0 and0
(I)

The complex roots are distributed in wide pairs, quartets and 2-strings as in [1]. The parameters of these roots verify a finite system of transcendental equations, similar to the original equations, up to a phase factor [eq. [25)]. The energy and m o m e n t u m of the excited states appear as superposition of contributions from elementary spin-½ excitations (holes). However, as we consider only N even, the total spin has to be an integer. As a consequence these spm-~ excitations always appear in pairs. In particular lowest excited states depend on two parameters. •

- 1
1

(II)

The complex roots are distributed as in I. Again the system of equations they verify is similar to the origonal one, but with a rescaling of the parameters [eq. (32)]. The excited states are superpositions of elementary spm-~ excitations (holes), but the condition that the total spin is integer yields a constraint between the number of holes, wide roots and close roots. This constraint implies that the holes always appear in pairs and that we reach only total spin zero states. The energy spectrum of these configurations is the one found in [4]. In the case where 3"/Ir is rational we can have states of non-zero spin. These rational points are then exceptional. A similar p h e n o m e n o n occurs in the 8-vertex model [5]* and sine-Gordon [6]. •

0 < A < 1.

1

(III)

We find two types of excited states: the first kind is a superposition of elementary * See also the last of refs. [2].

O. Babelon et al. / Bethe ansatz equations

15

spin-½ excitation (holes) and correspond to new multiplets of complex roots. The parameters of these multiplets again verify a finite system of equations of Bethe ansatz type, in fact the same as in case II. The condition on the total spin yields a constraint between the number of holes and the number of these multiplets: the holes appear in pairs. The total spin is again zero. The second kind of excitations can be viewed as bound states of two holes. They are described by strings. Notice that this arrangement of complex roots in strings appear naturally in the analysis and is not assumed a priori. 2. Bethe ansatz in the quantum inverse scattering approach To construct the eigenvectors and eigenvalues of the Heisenberg hamiltonian (1), it is convenient to use the quantum inverse scattering formalism [2]. Consider the local matrices t11(0) = (10

t22(O)=(W20 O)

W20(0))' 0

/'

01),

'

with

wx(O)

sh n sh(7 +0)'

w2(O) =

sh (0) sh(0+n)'

From these 2 × 2 matrices, we construct the 2 N x 2 u matrices Tii(0), i,/" = 1, 2, with elements

r T,i (o)] ~',:[:~,~. =

[tiq(o)]~rtq,z(o)]~...[t,~_H(o)]~.

Z il'"iN_ 1

Let us introduce the transfer matrix 2

~ ( o ) = Y. ~,(o). i-1

The elements of the matrices tii(O) are the local weights of the six-vertex model, and ~(0) is the transfer matrix of this model. The X X Z hamiltonian (1) is related to c~(0) through H = -sh n

0 log c~(0) ~

o=o

- N ch r/,

A = - c h r/. The m o n o d r o m y matrix [T(8)]0- = T~i(O) obeys the fundamental relation R ( 8 - 8')T(8) ® T(8') -- T(O') ® T ( 8 ) R ( 8 - 0 ' ) ,

(2)

16

O. Babelon et al. / Bethe ansatz equations

where [T(O) ® T(O')]~ = T~i(O)Tk,(O').

R(0)=

l WE(0) WI(0) W2(0) t" Wa(0) 1

Eq. (2) follows from an identical relation for the local matrices tij(O). Taking the trace in (2) we get

[~(o), ~(o')]

=

0,

vo, o'.

Hence log c4(0) is the generating functional for commuting conserved quantities. Diagonalizing H is the same problem as diagonalizing cO(0). In order to diagonalize c¢(0) we use the algebraic relation (2). More explicitly we have

(A(O) C(O)~ T(O)=kB(O)

D(O)]'

1 A(O)B(O') = w 2 ( O ' - 0 ) B(O')A(O)

wx(O'-O) B(O)A(O, ) w2(O'-O)

(3)

1 w~(O D(O)B(O') = w 2 ( O - O ' ) B(O')D(O) w 2 ( O -

B(O)D(O')

[B (0), B(0')] = 0 . In the quantum inverse scattering method, the next step is to find a reference state/2, eigenstate of A (0) and D(O) and annihilated by C(O). In the present model

.o=11)®ll)®...®11). where 1 denotes the vector of components (~). We have

A(O)[a) =

[a),

O(0)[O) =

wY(O)[a),

C(O)la) = o. We look for eigenstates of ~(O) of the form ~b(O~ . . . Op) = B(O~) . • • B(Opln>

(algebraic Bethe ansatz [2]).

(4)

17

O. B a b e l o n et al. / B e t h e a n s a t z e q u a t i o n s

Using the c o m m u t a t i o n relations (3) and the p r o p e r t i e s (4) of the r e f e r e n c e state we get A ( O ) O ( 0 1 . " Op) = A(O, 0 1 ' "

0p)~/(01'"'

Op)

P

+ E A,(O, 0 1 " "

O,)O(O, 0 1 " "

i=1

D ( O ) O ( 0 1 " " Op)=A'(O, 0 1 " "

0~'"

Op),

O p ) l # ( 0 1 " " " Op)

p

+ Y. A~(O, 0 1 " i=1

• Op)O(O, 0 1 " "

O, . . . Op),

where 0(0, 0 1 . . . 0 ~ . . . Op)= I] B ( O ~ ) B ( O ) I O ) ,

A(o, ol...

p o p ) = I-I

1 W 2 ( O i -- 0 ) '

i=i

p 1 A'(O, 01" " Op) = w ~ ( O ) ~=lrI w2(O - o~) '

w l ( O - O i ) F[ Ai(O, 01 " ' " Op)

A~(O, 0 1

" " " Op)

1

W2(O__Oi) l ~ i W 2 ( O j _ _ O i )

wl(O-Oi)

~ w 2 2(

--

N.

i)

,

1

,

(o,) I-[

]#i W 2 ( O i -- Oj)

T h e vector 0(01 • • • Op) is an eigenvector of c8(0) if A~(O, 01 • • • Op)+ A~(O, 01 • • • 0p) = 0 ,

Vi=I ..... p ;

this gives p equations for the variables 01 • • • 0p:

sh(Oi) ]N= l_i Sh (O~-Oj-n) Sh- (b~--r/)]

(5)

;~', sh ( 0 , - O i + T l ) '

and the eigenvalues are given by p

A(O, 01 • • • O p ) + A ' ( 8 , 0 1 " " " Op) = H

sh

i=1

(o,

+ rl)

sh ( 0 i - 0 )

+(sh(_o) .)N fi sb \sh(0+7/)/

i=1

sh(0i-0)

When A<-I we set r / = y ~ R and O j = i v j - ½ y . T h e n eq. (5) reads as eq. (6). 1. W h e n - 1 < A < 1 we set in eq. (5) r / = iy, 3' ~ [0, zr] and 0j = vi - ~ t y ; we thus get eq. (28).

18

0. Babelon

CaseI:

A<-1.

(a) Bethe ansah

et al. / Bethe ansatz equations

WesetA=-thy equations.

(O
ansatz equations

are [2]

(6) we suppose (Re ~1)c $7~.In the rational limit ui + 0, y + 0, Vi/r finite we recover the system studied in [ 11. We denote by Ai, i = 1, . . . , M, the real roots and by zr = al + i~[ the complex ones. In appendix A, we prove that in the N + 00 limit, if zI is a root then fl is also a root. We shall choose 71> 0. The system (1) splits in two parts, by specifying uj real or complex, Sin(hj+&)

N=_

sin (Aj -ii-y) >

sin (Aj - ZI+ iy) M sin (Aj -hi + iy) n n i=rsin(Aj-Ai-iy) I sin(Aj-zr-iy) x sin (Aj - fr + iy) (7)

sin (Aj - fr - iy) ’ sin (Zj + $7)

N =_

sin (Zj - +iy)

fi sin(zj-Ai+iy),sin(zj-zr+i,) I sin (zj-zr-iy)

i=r sin (Zj-Ai-iy) xsin(Zj-Fr+iy)

(8)

sin (Zj - & - iy) ’ We first solve eq. (7) in the thermodynamic limit (N + 00). (6) Integral equation. We take the logarithm of eq. (7) to get N4(Aj,ty)=

c” 4(Aj_Ai, i=l

Y)+C(4(Aj-z,, 1

Y)+4(Aj_fl,Y))+2irli,

(9)

where sin (z +icu)

(

4(zy”)=log where the determination

sin(z-icu)

of the logarithm

>’

is taken such that 4 (z, a) is a continuous

function for real z = x E )-&r, +$r(. Then 4 (x, (w) is a monotically decreasing function and we choose 4 (0, (Y) = ir. The 1j are half-odd integers. When N + co, the Aj tends to have a continuous distribution with density d(Aj)=

1 lim N-m N(Aj+r - Aj)’

(-ir
For simplicity of notation we shall use the density state the integers 1j form a monotonic sequence lj+l

-Ij

=

-1 .

p(A) = Nd(A).

For the ground

o. Babelon et al. / Bethe ansatz equations

19

For excited states the s e q u e n c e / j exhibits jumps for some values of/': N h

It+l-Ij =-1-

Y. 8j,j~. h=l

Taking the difference between eq. (9) for/' = k + 1 a n d j = k, we get in the limit N -->c~ +rr/2

f

d l x P ( l x ) f b ' ( A - t t , 3")=2i~p(A)+Nc~'(A,~3")-~.(q~'(A-zt, 3")+qb'(A-21,3"))

.,t--~12

l N h

+2i~r Y~ ~(~--~h),

(11)

h=l

where we have used +0/2

lim ~ f(Ak)

N~c~ k = l

f

~" J - w / 2

f(A)p(A)dA,

0

4,'(X, 3')= ~-~(X, 3"). /k+l --/k lira ~ ¢ ~ N(Ak+I-- Ak)

NI______ P(/~)--N

1 Nh E a(/~--Oh)" h=l

From this relation one sees that each jump in the sequence /j corresponds to removing a root at the point Aj, = Oh. One says that there is a hole at the position ~h (h = 1 . . . . . Nh). Eq. (11) can be solved by using Fourier series expansions -I-cx3

p(A)=

~

f i ( m ) e 2imx,

tel =--oo q-oo

4~'(x +iy, 3")=

~,

c , , ( y ) e 2imx,

m = --oo

cm(y) = - 2 i sign (y +3") exp (-2]m (y + Y)I) • The expression for c.,(y) leads us to distinguish two types of complex roots: close roots for zt < 3/and wide roots for ~'t > 3'. The solution reads P ( ~ ) = Pvac(}[ ) "~-Pholes(/~ ) + Pclose(/~ ) + Pwide(X ) .

The Fourier coefficients are given by dvac(m) =

fihole~(m)-

N 27r ch ( m y ) ' 1

2~r ch m y

N.

elmvl ~ e -2i"°h , h=l (12)

20

O. Babelon et al. / Bethe ansatz equations

1 /gclose(m) --

27r ch my 1

/~wide(m) =

el,,,vl ~ e-Zi,-o-~ (e -2Ira f~v-~'~) + e -21,. I(v+~- ))

c=1

Nw

el"/[ 2

2zr ch m y

e-Zim°'w(--e-21mp('%-V)+e-21ml(v+~'w)) •

w=a

Using the Poisson r e s u m m a t i o n formula, we get N

+oo E

Pac(Z) = m

=--oo

1

(13)

ch [(rr/y)(z + mrr)] "

F r o m the definition of p (it) we have for N ~ 0o +w/2

M

f

p(it)dit = ~'t3(0) = ~1N - ~ N 1h - N ~ .

a-Tr/2

Since N is even and M must be an integer, Nh must be even. It is useful to introduce the regular density Nh

• 8(it--Oh)=pvac(it)+O'(it).

tr(it)=p(it)+

h=l

In terms of this density eq. (11) reads +~r/2

f

~rr/2

dtzo'(lx)qS'(it-tz, y)=2icro'(it)+F'(it),

(14)

where Nh F(it) = Z ~b(it --Oh, y)--Y~ (~b (it --Zl, y)+~b(it -z.t, y))+Nc~(it, ½y), h=l

l

F ' ( A ) = O F (it).

Remark : O n e must not forget that in all this analysis we have retained only the leading terms for the v a c u u m as well as for the excitations. W e thus disregarded corrections to the v a c u u m which could be of the same o r d e r as the contributions of the excitations. This can be d o n e because these corrections to the v a c u u m only describe finite size (surface) effects, and are thus irrelevant here. (c) Higher level equations. W e evaluate the p r o d u c t over the real roots which a p p e a r in eq. (8) using the density p (it), ~ sin(zi-it,+iy)

1-I =exp i= 1 Sm ( Z i - ~ii- iT)

(.j+=/2

-

=/z

)

p(it)cb(zj-it, y) dit ,

eq. (3) then reads exp (I (zj) - F(zj) ) = - 1 ,

"

O. B a b e l o n et al. / B e t h e a n s a t z equations

21

where +w/2

I ( z ) = fJ-,,/2 dit o'(it)d (z - i t , 3").

(16)

The problem is now to evaluate I ( z ) for a generic z. For real z the integral equation (14) gives directly the value of I'(z) = d I ( z ) / d z : I'(it) = 2i~ro'(it) + F'(it ).

(17)

We shall first study the analytic properties of I'(z) in the z-plane. We have t" + ~ / 2

I'(z) = ]

,L- ~ r / 2

it(it )¢'(z - it, 3") dit,

with - i sh 23' ¢'(z, 3') - s i n (z +i3,) sin (z -i3") " We see that I'(z) is analytic in each of the three regions of the z-plane:

Ilmzl<3",

I m z >3",

Imz <-3".

Since o'(z) and F ' ( z ) are meromorphic functions, eq. (17) implies that

I'(z) = 2i~'tr(z) + F ' ( z ) ,

for IIm zl < 3".

(18)

When Im z becomes larger than % the pole at it = z - i 3 ' of ¢ ' ( z - i t , 3") crosses the contour of integration in the it-plane. By the residue theorem

I'(z)=2iTr(tr(z)+cr(z-i3"))+F'(z),

when Im z >3".

(19)

I'(z) =2irr(o'(z)+cr(z +i3"))+F'(z),

when Im z < - 3 ' .

(20)

Similarly

Notice that eqs. (18)-(20) give some information about the analyticity of the three functions 2hro-(z), 2iTr(o'(z)+o'(z-i3")) and 2iTr(tr(z)+o'(z+i3")) in the regions IIm zl < 3', Im z > 3" and Im z < - % respectively. Since I'(z) is analytic in these three regions, the poles of these (three) functions are the poles of - F ' ( z ) in the appropriate regions. They are located at z = Oh +i3" (residue +1), z = zt +i3" (residue q: 1), z = 2t + i3' (residue ~: 1) and + ½i3" (residue + N). We first evaluate I ( z ) in the region Im z > 3'. Remarkably we have

2izr (o"(z ) + o"(z - iN)) = G'(z ), where Nh

G(z)=-

1,

Nw

Y. ¢(Z--Oh--~3',½3')+ h=l .q_ S" e./~ ( Z

¢ ( z - z ~ , 3") + ~b(z - 2 ~ -i3', 3") w=l

1. 1 1. - z ~ -~t3", ~3') + 4,(z -z:~ -~t3,, 13,).

O. Babelon et al. / Bethe ansatz equations

22

This can be easily established by using the explicit f o r m of the Fourier coefficients (12). Notice that

Pvac(Z) "l-Pvac(Z - i v ) = 0 . Noticing that I(ioo) = 23, f~-/2 J-,rr/2 or(A) dA = G(ioo)+F(ioo), we m a y evaluate I ( z ) by integrating eq. (14); we get:

I(z)=G(z)+F(z),

Imz >3,.

H e n c e the equation for the wide roots reads exp G(zw) = - 1 .

(21)

In o r d e r to determine I ( z ) for IIm z I < 3,, we must evaluate its discontinuity on the line I m z = 3,. W e have

I (x +i3, + i O ) - I (x +i3, - iO) = 2i~r I or(z -i3,) dz , where the c o n t o u r is shown on fig. 1. Finally we get

I(z)=2iTr

L

Or(u)du+F(z)+G(a),

Ilmzl<3,.

It must be n o t e d that a change on the c o n t o u r joining a and z adds to I ( z ) an integer multiple of 2i~r. This does not affect eq. (15). In the region IIm z l < 3, we can write eq. (15) as

(

exp \2i¢r

du +

= -1

or

(

exp \ 2 i l r

0vac(U) du +2i~r

6"(u) du + G(a) = - 1 .

(22)

W e want to solve eq. (22) in the limit N ~ oo, while the first term is p r o p o r t i o n a l to N and the s e c o n d term has no explicit N d e p e n d e n c e .

x + i'y

I a = 1r/2 + i~"

I b

Fig. 1.

O. Babelon et al. / Bethe ansatz equations

23

Using eq. (13) we see that +co

Re (2iTr ff+~Ypv~c(u) du) =½N

E ....

log ch [(Irl3")(x + m ¢ r ) ] - s i n (,ryl3") ch [(~'13")(x + mrr)] + sin (zryl3")'

which is negative for 0 < I m z < 3'. F o r the two parts of the e q u a t i o n to match, the point z has to a p p r o a c h to order --otN e a pole of 2i~-oV(u) with negative residue. T h e r e is only one type of such pole in the region 0 < I m z < 3/, located at z = ~c + i3'. T h e r e f o r e if z is any close r o o t (0 < I m z < 3"), it has to have a p a r t n e r ~ +i3', also a close root. 1. 1. If I m z = ½3", then z is a m e m b e r of a 2-string (x + ~t3', x - ~t3"). If I m z # ½3" then z is a m e m b e r of a quartet (z, $, ~?+ i3,, z - i3'). By multiplying eq. (22) for the m e m b e r s zc and zc - i3, of a quartet or a 2-string the N d e p e n d e n c e drops out and we get exp (G(zc) +6) = - 1 ,

(23)

where a

6 = G(a)-2iTr

i

o'(u) du.

a-i~

Notice that 6 is i n d e p e n d e n t of N since ~_ivpvac(U)du = 0. M o r e o v e r , for the origin of the minus sign in the r.h.s, of eq, (23) see appendix B. Let us define 1.

X = z --2t3'

for a 2 string (z, z -i3"),

X = ~:+ i(z-½3")

for a quartet (z = ~ + i~', ~, z - i3", ~ + i3"),

x

for a wide pair (( + i~-, ~c_ i~-).

W e thus get a set of p a r a m e t e r s

{X1 . . . . .

(24)

X~}

= no. strings + 2 • no. quartet + 2 • no wide pairs. Eqs. (21) and (23) then take the particularly simple f o r m N. sin (Xi -Oh + 213") _e~8 sin O(i - X i + i3") h=l sm (Xj--Oh ½i3")i=1 sin (Xj--X~--i3") ' where e i

----

0 if Xi is the p a r a m e t e r of a wide pair, e i

6 -

=

(25)

1 otherwise:

log c°--~s 2 0?i - ½i3") j=l

COS

(Xi+~i3")

+ X (--2iOh + log ~ c°s2 (8~ + (2r + 1)i3") cOS2 (Oh -- (2r + 2)i3")~ h=l

r = 0 COS 2 (Oh --

(2r + 1)i3')

COS 2 (O h +

(2r + 2)i3")]

"

Notice that these equations r e d u c e to the ones f o u n d in [1] in the rational limit.

24

O. Babelon et al. / Bethe ansatz equations

(d) Energy momentum and spin. The energy of a configuration above the ground state is given by [2]

--//+=,,

,

'

,_

q

zIE = - i sh "YLa-=/E dr(u)d>'(u, 13/)+ 2 ¢b'(zu 2y) +¢b (zt, ½y)j The momentum operator P is defined from the unit translation T as 7" = e -ie and the m o m e n t u m of a configuration above the ground state is given by

A P = - i t . j _ ~ / 2 o'(u)qb(u,~y)

~.q~(Zt, ½y)+c~(~t,

.

These integrals can be evaluated explicitly: neither the wide pairs nor the quartets or the 2-strings contribute to AE and zIP. The only contribution comes from the holes and N,

zIE= Y e(0.), h=l Nh

zIP = Y~ p (Oh), h=l

where shy 2Ksh (2KO, k ) , e (0) = 2~- --~-Ov,c(O) = ---~- y d n \ - - ~ -

(26)

[2KO ) 1 p(O ) = am~--~--, k - ~Tr .

(27)

pvac(O) is the ground-state density given by (13) and K and K ' are the complete elliptic integrals of modulus k and ~ respectively, such that K ' / K = y/r:. The dispersion relation follows by eliminating 0[7, 4] from eqs. (26) and (27): e (p) = 2---Ksh yx/1 - k 2 cos 2 p . 'W

Notice that there is a mass gap for y ~ 0:

e ( p ) > e ( O ) = 2 K s h y~/]----------~. "tt

Since the spin operator N

s =½y i=1

commutes with H, we may evaluate it on the eigenvectors of H. It is given by ~N minus the total number p of roots of eq. (6):

Sz = ½N - M - 2Nc - 2Nw = ½Nh - N o - 2Nw . This takes only integer values since Nh is even.

O. Babelonet al. / Betheansatz equations

25

Case II: -1
(28)

W e suppose IIm v I -< 1 ~ ~Tr. T h e analysis of these equations is similar to the previous one and leads to the integral e q u a t i o n -boo

d/xp(/.t)~b'(A - i t , 3') = 2iTrp(A)+Nfb'(A, 13,)

f_ Oo

-Y. (~b'(A - z l , T)+d~'(A - £ t , y ) ) + 2 i r r l

/% ~ 6(A --Oh), h=l

with • sh (z + ice) 4~(z, a ) = log s - ~ - ice)" This e q u a t i o n is solved by F o u r i e r t r a n s f o r m ~ +o0

p(A) = ~ - j_~ fi(k) e 'kx d k , 1 "j~+oo

r~'(x, a) =~-£~

C(k, a) e ikx dk,

(x, ce r e a l ) ,

Oo

C(k, ce)=-2irrsign(ce)

s h k (~" 1 -Ice sh k ~ r

f)

'

- z r
1

W e distinguish wide roots ( y < I m z < ~ ) and close roots ( l l m z l < y ) . T h e density p (A) of real roots is defined as (10). T h e Fourier t r a n s f o r m s are given by N rivet(k) = 2 ch ½ky /3holes(k) = -

N, E e ikOh h= 1

A

k

O~oso( ) = - Z

fiwide(k) = 2

2

e - iktrc

c~t

sh Ik'rr 1

ch ~ky sh I k (zr -

y)

sh k (1or - y) ch krc i ~ ch~kysh~k(Tr-y)

,

Nw

~. e-ik"wsh½kychk(½7r-rw)l w=l sh ~k(Tr - y)

T h e integer M is n o w given by -bOO

M=~

Oo

p(A)dA=fi(O)=~N-2Nc-2Nw-ce(1Nh-Nc-2N~),

w h e r e ce = zr/(Tr-3/). W e shall i m p o s e 1Nh-N~-2Nw = 0 so that M is an integer.

O. Babelon et al. / Bethe ansatz equations

26

Notice that if v/~r is a rational number we can have a less restrictive constraint. In order to get higher level equations, we must evaluate the integral given by (16). We again have to distinguish three regions in the z-plane. For y
1

I' (z ) = 2izr (o"(z ) + tr (z - iv)) + F ' ( z ),

<~zr,

Ilmzl
I'(z ) = 2ilrcr(z ) + F ' ( z ) ,

1

-~"
(29)

I'(z ) = 2i~r (tr (z) + tr(z + iv)) + F'(z ) ,

where o'(z) and F ' ( z ) are defined as above. We have

2ilr(cr(z ) + o'(z - i v ) ) = G'(z ) , with N h

O(z)=- Z

4, [,~ (z - oh - ~iv), ~,~v]

h=l

1. 1 1. + Z 4,[~(z -zc -~v), ~o~v]+ 4,[~(z -z~ -~v), ½o~v]

¢=ql

N w

+ Y~ 4,[a(z - z ~ ) , o~VJ+4,[a(z -~w - i v ) , aV].

(30)

w=l

Noticing that d-OO

I(+oo+iy) =2iv l -

or(A) dA =2iv(½N + N h - 2 N c - 2 N w ) , co

we may evaluate I ( z ) by integrating eqs. (29). For v
1 <~zr,

I(z)=F(z)+G(z), (31)

I(z)=2i~r

[Imz]
tr(u)du+F(z).

Here again a change in the contour of integration will affect I ( z ) only by a multiple of 2izr. From eqs. (15), (31) were are led to the same analysis for the close roots: they appear again in 2-strings or quartets. In contrast to the case A < - 1 , the higher level equations for the wide roots and for the top member of quartets or strings are identical and read exp G ( z ) = - 1 , where G ( z ) is given by (30). Defining the set of parameters {gi} as in (24) the equations read N h

rr

1.

sh a(Xj --Oh +~iy)

sh a(Xj -X____2+iy)

sh, j-oh- iv These equations again reduce to the ones of ref. [1] in the rational limit.

(32)

27

O. B a b e l o n et al. / B e t h e a n s a t z equations

(b) Energy momentum and spin. Energy and m o m e n t u m above the ground state are given by +~

zIE=-isiny

d/x#(/x)¢ (/.t, g y ) + ~ ¢ co

zIP=-i

(i

(zl, 5-y) ¢ (zt,~y

,

(33)

l

1+

d/.t e(/~)O (~, ~y) oo

X&(zt, l y ) + ¢ ( 2 t , ~ y ) .

(34)

l

Again only holes contribute to energy and momentum: Nh

A E = Z e(Oh), h=l

Nh zIP = Z P (Oh), h=l

with e(0) -

~r sin y

(35)

y ch (frO/y) '

p(O)=2arctge °'~/~,

O<-p<-Tr.

(36)

The dispersion law is rr sin y . e (p) = - smp. Y

(37)

The spin is given by

Sz = ~ - M - 2 N c - 2 N w

=0,

where we have used the constraint ½Nh - N o - 2 N w --0. The first excitations corresponds to Nh = 2, Nc = 1, Nw = 0; that is to say two holes and a 2-string. These configurations are the ones found in [4]. When the roots ),j are distributed in a finite interval (d < - 1 ) we find states of non-zero spin. In the present case the Aj are unbounded and we only find states with S~ = 0. Since this constraint comes from imposing that S+~ p (A) dA is an integer, one may think that a better treatment of very large roots (where p (A) ~< 1/N) should help to understand states with Sz # 0. Case III: 0 < A < 1. We set A = - c o s y, lzr < 3' < zr. The Bethe ansatz equations are the same as (28) and the integral equation is solved by Fourier transform. In this range of values of 3' the solution is /gvac(k)

-

2

~ho,es(k) = -

N ch ~ k ~ ' Nh E e -ik°h k=l

sh lkrr 2 sh ½k(rr - y) ch ½ky'

O. Babelon et al. / Bethe ansatz equations Nc

1

~close(k) = - Y~ e -ik'° sh k(~rr - y ) ch krc c=1 ch ½ky sh ½ k ( z r - y) ' ~- -/k,, c h ½ k ( ~ ' - y ) ch k ( ~1 r - z w ) t~wiae(k) = - 2 L,we w ch ½k3, ' where by close roots we m e a n IIm z~l < ~ r - y and by wide roots we m e a n 1 r - y < IIm z I < ½~'T h e integer M is given by +OO

M=I_

p(A)dA=½N-2N~-2Nw+a(Nc-½Nh). oO

H e n c e we must have

Sc-½Nh =0, unless y/~r is rational. In the evaluation of I ' ( z ) , we must consider three regions in the z-plane: 7r - 3 ' < I m z <½zr,

I'(z) = 2i~r (tr(z) - t r ( z -i~r + i y ) ) + F ' ( z ) ,

Jim z I < zr - 3~,

I'(z) = 2izro-(z) + F ' ( z ) ,

-½~r < I m z < - ~ - + y ,

I'(z)=2izr(tr(z)-tr(z+iTr-iy))+F'(z).

W e m a y c o m p u t e I ( z ) by using I(~> + iy) = 2i (Y - ~r) ~_+~o'(A) dA. W e get

I(z)=2i~r

(o'(u)-o'(u-i~r+iy))du+F(z),

~ ' - y < I m z <½1r,

I(z)=2i~r

o'(u)du+F(z),

IIm zl < r r - y ,

I ( z ) = 2i~r

(o'(u) -o'(u +iTr - i y ) ) du + F ( z ) ,

-½rr < I m z < 3' - ~r.

T h e equations for the c o m p l e x roots are exp

2i¢r

(rr(u)-o'(u-iTr+iy))du =-1,

(

exp k2i~r exp (\2irr

o'(u) du

=-1,

¢r-y
(38)

lira z ] < r r - ~ , , (39)

(cr(u)-o~(u +izr

du = - 1 ,

- ~1- < I m z

Analysis of the Bethe ansatz equations of the XXZ model

Analysis of the Bethe ansatz equations of the XXZ model

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