Conservation laws for Z(N) symmetric quantum spin models and their exact ground state energies

Nuclear Physics B275 [FS17] (1986) 436-458 North-Holland, Amsterdam

CONSERVATION LAWS FOR Z ( N ) SYMMETRIC QUANTUM SPIN M O D E L S AND THEIR EXACT GROUND STATE ENERGIES Francisco C. ALCARAZ*

Department of Mathematics, The Faculties, Australian National University, Canberra A C T 2600 Australia

A. LIMA SANTOS

Departamento de F{siea, Universidade Estadual Paulista, "Julio de Mesquita Filko", 13500 Rio Claro-SP-Brasil Received 11 June 1986

We derive an infinite set of conserved charges for some Z(N) symmetric quantum spin models by constructing their Lax pairs. These models correspond to the Potts model, Ashkin-Teller model and the particular set of self-dual Z(N) models solved by Fateev and Zamolodchikov [6]. The exact ground state energy for this last family of hamiltonians is also presented.

1. Introduction

Most exactly solved models in two dimensions are known to be related to the eight-vertex model, solved by Baxter [1]. This model exhibits an infinite number of conserved integrals commuting with each other [2]. The generator of those integrals of motion is the row-to-row transfer matrix [1,2]. The conserved quantities are quantum hamiltonians describing different one-dimensional systems, in particular one of these describes the XYZ-model. Bashilov and Pokrovsky [3] also showed that an infinite set of conserved quantities can be obtained for the hamiltonian version of the N-state Ports model, at the critical point. In this paper we show that in this case the generating function of conserved integrals is the diagonal-to-diagonal transfer matrix of the classical N-state Potts model [4]. In this paper our aim is to find the possible models with Z(N) global symmetry that also show an infinite set of conserved laws. Starting with a general one-dimensional quantum Z(N) hamiltonian we try to construct a Lax pair L - A which permits us to derive directly the conserved integrals of motion [3]. Beyond the * Permanent address: Departament de F~sica, Universidade Federal de S~o Carlos CP616, 13560 S~o Carlos-SP-Brasil. 0550-3213/86/$03.50 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

F.C. Alcaraz, A. Lima Santos / Z(N) symmetric quantum spin models

437

solution of the quantum N-Ports model we are able to construct those L - A pairs for the quantum Ashkin-Teller model [5] and for a special family of Z ( N ) self-dual models. In all cases the generating function for the charges is the diagonal-to diagonal transfer matrix of the corresponding two-dimensional Z ( N ) classical model. This paper is organized as follows. In sect. 2 we introduce the general classical Z ( N ) model and derive its transfer matrix. In sect. 3 the corresponding general hamiltonian is introduced. The Lax equations and the solutions we obtained are given in sects. 4 and 5. Identifying the classical model [6] corresponding to our hamiltonians we are able to derive in sect. 6 the ground state energy for a particular set of Z ( N ) quantum hamiltonians. In appendices A and C we show the L - A pair for N = 4 and 5 respectively, while in appendix B we prove the involutivity of the integrals of motion for the Z(4) model. Finally in sect. 6 we present our summary and conclusions.

2. General Z ( N ) model - transfer matrix

Our Z ( N ) model is defined on a two-dimensional square lattice with sites labelled by the indices i, j (see fig. la). On each lattice point there is a spin variable S(i, j) satisfying

Is(i, j ) ] N= I.

(2.1)

We may equivalently introduce integer-valued variables n(i, j) for each site:

S(i, j ) = exp[~-n(i,i21r j ) ] ,

n ( i , j ) = 0 , 1 , 2 ..... N - 1 .

(2.2)

The general Z(N)-invariant model with next-nearest iterations we shall consider is I

I" I I -,I,.

I l I I

-,I, I

I

I' . . . . I I

I

I/

I/

I/

-'I I

I I

//

I I

//

/

/

I

/

I k

-/

~ .....

I I I I

I I I I

,I,.

4,-

I

I

Fig. l(a)

,, } Y

/ I I I

~//

//I /

/

/

/"1 I

I ~//

/

~

/

/

I

I I

..i,/ /I

/I Fig. l(b)

Fig. 1. (a) Normal square lattice, (b) deformed lattice.

/I

F.C. Ah'araz, A. l.ima Santos / Z(N) symmetric quantum spin models

438

defined by the hamiltonian [7]

H = - _.21½Jxl[S(i' j ) S + ( i + 1, j) + c.c.- 2] I,J

+ ½Jx2[(S(i, j)S+(i + 1, j))2 + c . c . - 2] + .." +½Jx;[(S(i, j)S+(i+ 1, j ) ) ~ 7 + c . c . _ 2] 1 [S(i,j)S+(i, J + 1) + c.c. - 2] + ~Jrl +½Jy2[(S(i, j)S+(i, +'"

j+

1)) 2+ c . c . - 2] 2

+ ½Jy~[( S(i, j)S+(i, j + 1)) ~7+ c . c . - 2]},

(2.3)

where c.c. is the complex conjugate, N is the integer part of IN and Jxi, JYi; i = 1,2 . . . . . .N are the coupling constants in X and Y directions respectively. In terms of the integer variables introduced in (2.2) this hamiltonian can also be written as

H=-~

Jx,~ cos ~ - a ( n ( i , j ) - n ( i + l , j ) )

-I

~,i ~ = 1

~ - a ( n ( i , j) - n(i, j

-

}

(2.4)

In order to obtain the diagonal-to-diagonal transfer matrix it is convenient to deform the square lattice as in fig. lb. The transfer matrix T is an operator acting on a Hilbert space ~ h as follows. Take the vertical direction in fig. lb as the "time" axis, then at a particular time t = t 0, the state of the system is described by a vector in the direct product Hilbert space fit h spanned by M/2

I~ In(j)),

In)= j=

. M/2

where in(j)) describes the spin state at site i at t = t o and M is the lattice size in the horizontal (space) direction. The dimension of !fib is then N M. By choosing In(i)) as the eigenstates of the unitary operator S(i):

S(j)ln(j))

[ i2~ j))ln( j)).

= exp[ ~ - n (

(2.5)

F.C. A Icaraz, A. l.ima Santos / Z(N) symmetric quantum spin models

439

We define T as the operator whose matrix elements are e x p ( - f i l l ) :


(2.6)

where n = {hi}, n ' = {nS} are the state vectors of two adjacent rows (see fig. lb). In order to express (2.6) as a matrix element of some operator, we introduce rotation operators J~(j), which rotates the spin j by an angle 2rr/N

R(j)ln(j))

= I n ( j ) + 1)

(2.7)

(mod N ) ,

those operators satisfy the Z(N) algebra: 5~N = ~m = 1,

[ S ( i ) , S ( j ) ] = 0, [ S ( i ) , / ~ ( j ) ] = 0,

[/~(i), R ( j ) ] = 0,

(2.8a)

i 4=j,

(2.8b)

R( i )S( i ) = ei2"/NS~(i ) R( i ) .

(2.8c)

In terms of these operators the transfer matrix is expressed as:

M/2 TD=

l-I i= - M / 2

Tl(i)T2(i+

1),

(2.9a)

where

Tl(i ) = e x p

E

~Jx,~[(g(i)S+( i + l)) " +

h.c.- 2

,

(2.9b)

a=l

T2(i ) =

~

h~(i)exp

~

Blv,~ cos

-1

.

(2.9c)

To see that this form is correct it is convenient to define the Boltzmann factor in the X-direction

r/=0,1 ..... N-l,

(2.10a)

440

F.C. A lcara:, A. Lima Santos / Z(N) sTmtnetric quantum spin nlodels n1-1

n2+2

.4

/ I I

/

n3

,,4'

/

/

I

/

I I

/

n4+1

/

/~

I I

/

/

/

I I

/

I//

1/ /

I / /

I /

,V

,V

i,/

~/

nl

n2

n3

n4

Fig. 2(a) n1-1

n2+2

•

n3

•

I I[ Y{-1)R+(1) ^

I IIY{2)R^ 2 (2)

I

•

I

I y(o)~O(3 )

I

--~--71(1) nl

n4+1

•

I

~-n2

T1(2)

4-n3

I

^ IY(1)R(4)

I T1{3)

4-n4

Fig 2(b)

///D n1

n2

n3

n4

Fig. 2(c)

Fig. 2. See the text. (a) A particular configuration of two adjacent rows for the lattice of fig. lb. (h} The operators that connect the two rows of fig. 2a. (c) The matrix element of TD between the two configurations in fig. 2a.

and Y-direction

Y(v/)=exp

{N i

Y2/3Jy~ cos ~ - c ~ a=l L \

) 1) -1

,

~=0,1...N-1.

(2,10b)

Consider n o w the two configurations {n(i)} and {n'(i)} shown in fig. 2a. The matrix element ( n ' I T I n ) between these two rows will involve the operators in fig. 2b. The matrix element will be the product of the factors in fig. 2c, where the diagonal interactions appear because the flipping operators /~(i) act before the c o r r e s p o n d i n g S + ( i - 1)S(i) operators. We see therefore that (2.9) corresponds to the diagonal-to-diagonal transfer matrix. The order of the product of factors T 1 and T 2 in (2.9) is very important, for example, it is simple to see that the row-to-row transfer matrix T R is given by [7]:

TR=

M/2 I-I [Tl(i)T2(i)], i 1¢/2 =

(2.11)

F.C. Ah'araz, A. Lima Santos / Z(N) symmetric quantum spin models

441

where T 1 and T2 are given in (2.9b) and (2.9c). For our future analysis it is convenient to express Tl(i) as a sum of operators: Tl(i) = ~

[S(i)S+(i+ 1)]"X(n),

E

(2.12)

~1=0

where X(,/) is the dual (Fourier transform in the Z ( N ) group) of the Boltzmann weights X(,/):

)((rl) =

a)exp

arl

E X(a), a

7=0,1 ..... N-1.

(2.13)

=0

We can express the transfer matrices in a more symmetric form by doubling the size of the original lattice and introducing a new set of operators p(i) as

p(-2i)=R(i), p(-2i-

1) = S+(i)S(i + 1),

i = - ~M . . . . . [~M.

(2.14)

From the relations (2.8) we can see that these operators obey the Z(N) algebra

oN(i) = 1,

[P(i), P ( j ) ] = 0,

Ii - j [ > 1,

p(i)p(i + 1) = eiZ"/Np(i + 1 ) p ( i ) .

(2.15a) (2.158)

The normalized diagonal-to-diagonal transfer matrix TD can then be written simply as

TD= TD

N a=o X(a)

-- a=[1+M Lk(hk),

(2.16a)

where N-

1

Lk(Xk)= ~ pn(k)hk(~)

(2.16b)

h,(~/) = Y(T/),

k even,

(2.16c)

2,k(~ ) = )((~),

k odd.

(2.16d)

and

In the infinite system by shifting the lattice by one unit we obtain the result

TD(X, Y) = TD(Y, X),

(2.17)

442

F.C. Alcaraz, A. l,ima Santos / Z(N) ,~vmmetric quantum spin models

where )?(-q) is given by (2.13) and I7(7) by a similar expression. The result (2.17) may also be obtained by performing a duality transformation in the original transfer matrix (2.9) [7]. A special region in parameter's space { X(v/), Y(v/)} is the self-dual region ~ s , composed of the self-dual points: X(v/) = Y(v/),

~=0,1...

N.

(2.18)

For example, for N = 2, 3, 9t s is a line while for N = 4, 5 9t s is a 4-dimensional hyperplane. In 9is the transfer matrix (16) has the simple form: M

TD =

H

Lk( X ) ,

(2.19)

k=+M

where L k is again given by (2.16b).

3. General Z(N ) quantum hamiltonian In order to obtain quantum models with an infinite set of conservation laws we introduce in this section the general self-dual Z ( N ) quantum model, whose dynamics are described by the hamiltonian: H(2t) = H, + XH 2 ,

(3.1a)

where M/2

Ht = -

N-

~ 1=-M/2 M/2

H2=-

(3.1b)

7=1 N

E i=

l

~ C T ( S + ( i ) S ( i + 1)) 7 , l

E Gk"(i),

c.=cN 7

(3.1c)

M / 2 ~1= 1

X plays the role of the temperature, C,; 7/= 1,2 . . . . . N are real coupling constants and the operators /?(i) and S(i) obey the algebra given in (2.8). The coupling constants are chosen to be real so that it should be possible to relate these hamiltonians to the transfer matrix introduced in the last section*. By introducing the 0 operators defined by eq. (2.14) we can rewrite the hamiltonian as

,(x)=

-

E i= -M

E Gp"(;),

(3.2)

*/=1

* Von Gehlen and Rittenberg[8] find particular models with an infinite set of conservedcharges where the coupling constants are complexnumbers.

F, C. A lcaraz, A. Lima Santos / Z(N) ~Tmmetric quantum spin models

443

where e i = X for odd sites and ei = 1 for even sites and the operator p ( i ) obeys the commutation relations given in (2.15). Again in terms of those variables the duality transformation for the infinite system is obtained just by shifting the lattice by one unit. Hence

In particular at the self-dual point X = 1, the hamiltonian has the simple form: M

H(1) = -

N

1

E

E

i=--M

~/=1

(3.4t

In the next section we will investigate the existence of points in the space of coupling constants {Co } where the above hamiltonian has an infinite number of conservation laws.

4. Lax pair equations The equation of motion for the operators Ok is given by -it5 k = [H, phi.

(4.1)

We will restrict ourselves hereafter to the self-dual hamiltonian (3.4), therefore the equations of motion of the operators are explicitly N-1

-it5]=-

~_,

Cpotn,p[pnkpf_l--p;+lpnk],

~/=1,2 ..... N-l,

(4.2a)

p=l

where an.p = exp( i2 ~r~ l p / N ) - 1.

(4.2b)

On the lattice we shall consider a Lax pair to be a pair of operators L k - A k obeying the relation [3]: - i l k = Ak+ 1Lk - L k A k ,

(4.3)

where L k, A k are the local functions of the operators Pk and some spectral parameter ~. If such pairs of operators exist then the quantity £P(Tk)=

f - ] Lk(~, )

(4.4)

444

F.C. A h'araz, A. l.ima Santos / Z(N) .~3'mmetric quantum spin models

is conserved. Moreover (4.4) will be a generator of charges; its coefficients in the power expansion in X being the motion integrals. Our approach is to determine for which coupling constants { Co } in (3.4) a pair of operators satisfying (4.3) can be found. The structure of the equations (4.2) and (4.3) induces us to try to find an L - A pair of the form: N

1

L~= E loo],

l , = l u o'

l0 = 1 ,

(4.5a)

=0 N

1 N

Ak = ~

1

~

i=0

ai, i P ik P kj

(4.5b)

l,

j=O

with

ao, o = 0 ,

a*i, i=

(4.5c)

ai, N ./~- a N _ i,.!

The Lax pair equation (4.3) indicates that the operator has an arbitrary constant which we chose to be zero in (4.5c). From (4.5a) and (4.2a) the left side of (4.3) is given by N-IN

-iL,=-

~ ,7=0

I

o p Z loCpan,pt[ p Ok p Pk-l--OkPk+l),

(4.6a)

p=l

where a p is given by (4.2b), while the right side is

Ak+lLk-LkAk

N

1N-1N

= Y',

2

i= 0

j = 0

1

~-, ai, il,,[O~+lO~"-oi~+"O~

1].

(4.6b)

n= 0

Forcing (4.6a) to be equal to (4.6b) the following equations should be satisfied N

!

Y" ai, ,!, = 0,

i = 0, 1 . . . . . N - 1,

(4.7a)

i = 1 , 2 . . . . . N - 1, m = 0 , 1 , 2 . . . . . N - I ,

(4.7b)

/=0

N

1

~_. a~./l{., i} j=o

=

lmCiam.i'

aij = aji,

(4.7c)

where { n } denotes n, modulo N. To proceed further it is better to decompose the

F C. A Icaraz, A. Lima Santos / Z(N)symmetric quantum spin models

445

a,, / and a,,j into their real and imaginary parts a , , / = A , , / + iB,,/,

(4.8)

i,j=O ..... N-1. ai, I= •i. ] -~- iT,, : ,

Eqs. (4.5c) and (4.7c) give us Bo, i - ~ B i , o = O ,

Bi,u-j Ai,j=Aj,

BN/2,1 -~ Bi, N/2 = O,

A i , g = Ai, N - j

= -Bi,j, Bi, j = Bj, i,

i,

Neven,

for

i, j = 0 . . . . . N -

1,

(4.9)

a n d the relation (4.7a) becomes N

1

Ai, o + Y', Ai.gl J = O, j=l

i = 0,1 . . . . . N - 1,

(4.10)

while (4.7b) gives us two equations N

1

(4.11a)

Ai,{./ ,,}l~ = l/CiB~, i, n=0 N-1

Y'. Bi.{: ~}1,7= l/C,Ti,:,

(4.11b)

n=0

where i = 1, 2 . . . . . N - 1 and j = 0,1 . . . . . N - 1. Eqs. (4.10)-(4.11) with the relations (4.9) are our central equations in the calculation of the L - A pair. In the next section we find possible solutions for the above equations.

5. Lax pair solutions A first class of solutions, for general N, which has been obtained in the literature [3], c o r r e s p o n d s to the q u a n t u m Potts model. In this case the coupling constants of the h a m i l t o n i a n (3.4) satisfy C,-= C ,

~ = 1..... N-

1.

(5.1)

A n L - A pair for this particular case, can be obtained by using the following structure for L and A 10 = 1,

l i = 2~, ai, j = a ( X ) +

i = 1..... Nj.

1,

(5.2) (5.3)

446

F.C. A h'araz, A. l.ima Santos / Z(N) symmetric quantum spin models

By using eqs. (5.2) and (5.3) in (4.7) we obtain

C)~ b(?~) -

CX2N

l-h'

a(X)-

I+X(N-1)'

(5.4)

which corresponds to that obtained by Bashilov and Pokrovsky [3]. Moreover comparing (4.5a) and (5.2) with (2.19) we see that the diagonal-to-diagonal transfer matrix is the generating function for the conserved charges. This connection enables one (see eqs. (5.9)-(5.10)) to use the exact results for the critical classical twodimensional Potts model [1] to rederive the ground state energy of the one-dimensional quantum Potts model at the critical point [9]. For N > 3 the hamiltonian (3.3) is more general than that of the Potts model. In the following we will try to find points in the space of coupling constants beyond the Potts point (5.1), for which we can define an L - A pair.

5.1. Z(4)

The Z(4) hamiltonian

-H= F [Cl(ok+O;)+GO]]=G Hk k

(5.5)

k

corresponds to the quantum version of the Ashkin-Teller model [5] at the self-dual line. This quantum hamiltonian is obtained by the extremely anisotropic time continuum limit [10] of the classical Ashkin-Teller model (N = 4 in (2.3)). The L - A pair, corresponding to this hamiltonian, is derived in appendix A, where we show that

Lk= l + X(pk + p~ ) + ~t

C2+ Clh C 1 + hC 2

p],

(5.6)

with the operator A given by eqs. (A.10). We find therefore that for any couplings C1, C 2 there exists an infinite set of conservation laws. The motion integrals Q, are given by the coefficients in the expansion in powers of h of the generator (4.4)

,£P()t)= i-~ Lk(X)= ~ Qnx"k=+~

(5.7)

n=0

The first motion integral is related to the hamiltonian

Q~= L Hk=-H/C~,

(5.8a)

F.C. A h'araz, A. Lima Santos /

Z ( N ) ,lvmmetric quantum ,Vml models

447

while the second and third are given by

£

Q2=

HkHk"+ (I - C2) £

k > k' =

Q3 =

vo

k=

£

V~,

(5.8b)

oc

HkHk'H~"+ (1 - C 2)

k>k'>k"

X

£ k>k'=

(HkVk,+V~H,,)-(I-C 2) £ ~c

k=

Vk,

(5.8c)

zo

where C2

Vk = O~,,

C = --. C1

(5.8d)

Apart from Q1 all the other motion integrals Qn are non-local. In order to obtain local conservation laws I n we should take the logarithm of the generating functional ~ ( X ) [3]. The expansion in powers of X of logL~°(X) will give us the local conservation laws: l o g ~ ( X ) = £ In?tn,

(5.9a)

n=l

where 1,= Y ' i n ( k ) .

(5.9b)

k

in(k ) is a function of Ok operators with no more than n points positioned successively and in(k + 1) is obtained from in(k ) by just a unit translation of all the operators. The first local charge is again related to the hamiltonian I, = ~H

/,

k =

- H/CI,

(5.10a)

while the second and third are given by:

I, = , ~Z;[H,,+I,H,,]+ k

(1 - c ' ) E v~ - ~Y'.H;, ' k k

(5.10b)

13 = E { ~ H ~ - C(1 - C2)Vk - ½(I - C 2) k

x (H,
+Hk[Hk, H k + , ] + [ H k , H k + l ] H k + , + [H2, H,+,] + [Hk, HE])}.

(5.1Oc)

448

F.C. Alcaraz, A. Lima Santos / Z(N) symmetric quantum s'pin models

We can also show that not only [H, I,] = 0 but that all the I, commute with each other, i.e. [,In, I,,] = 0 for arbitrary n and m. For this purpose it is sufficient to prove that for arbitrary 2~ and t* the two generating functions ~ ( X ) and X'(/,) commute: [&o(?~), .£#(/~)] = 0 .

(5.11)

This is done in appendix B. The fact that we find an infinite set of conservation laws for CJCt general may be seen as a consequence of the equivalence of the hamiltonian (5.5) to the anisotropic XYZ model [10]. In order to see this equivalence it is better to rewrite (5.5) in terms of our original variables R, S (see eq. (3.1)):

u=

--

E{[c,(~,+~7)+c2k~l+[c,(s,s,+,+s, ^^+ ^ + ^s,+~)+ C28i23i21]

}

(5.12)

i

Introducing two Pauli matrices o[, r[; oil ri ~ at each lattice point we can rewrite (5.12) as

/4= - E { [c,[o,x¢, x + , , q + c2o,.1 +

[c,(o,%, + o:o7+1,/,,~,)+ c2q,/+11 }.

i

(5.13) Performing a dual transformation only in the r variables

°j~+l/2=rfrf+l,

°f+1/2 = 1-I rxk

(5.14)

k _<.j

and doubling the lattice, (5.13) becomes: H = - 2 ~ { C 1[ o[o,+ 2 + o,:o,4+1Oi+2 ] Jr- C2oix+l } .

(5.15)

i

Now we do another dual transformation in all variables

S[+I/2 = 1-I o;,

Si+l/2 --- o[o,%t,

(5.16)

k<_i

and by doubling the lattice again we get

H = - 2 £ [ C, (S,~Si+,

-

Si~'S[V+l

) -1- C28787+

1],

(5.1v)

i

which is just the

XYZ model with one coupling fixed. Doing the same transforma-

F.C. A Icaraz, A. Lima Santos / Z(N) s3'mmetric quantum .spin models

449

tion for the L - A pair we get

L k ( X ) = 1 + X(S,"Si'+I - S"SY, ,+ t~~ + ~ C2 + C~X

C~ + XC2

(5.18)

and a corresponding expression for A k. This pair corresponds to that found in ref. [3] for the X Y Z model. The first calculation of the conservation laws of the X Y Z model was done by Liischer [2]. Although his L - A pair was constructed by using an auxiliary 2-dimensional space, it is possible [3] to relate them to that given by (5.18) and (A.10). T o finalize this section let us make the connection between our hamiltonian and the transfer matrix for the Ashkin-Teller model. By comparing (2.18) and (2.19) with (5.6) and (5.7) we see that the generator of the infinite set of conservation laws is the self-dual diagonal-to-diagonal transfer matrix of the Ashkin-Teller model with the Boltzmann weights introduced in (2.10) and (2.13) given by Xl = )~ = ~'l,

X2 = Y2 = ~k

C 2 -~- ~kCl

(5.19a)

(5.19b)

C t + XC 2 "

The same connection can be made between the X Y Z hamiltonian (5.17) and the row-to-row transfer matrix of the eight-vertex model [1]. As is well known [11] the Ashkin-Teller model may be mapped to a staggered eight-vertex model (see fig. 3) which becomes non-staggered and exactly soluble [1]

Fig. 3. Equivalence between the Ashkin-Tellerand the eight-vertexmodels. The first model is defined in the lattice L with circles in the vertices while the second one in the medial lattice of L.

F.C. Ah'araz, A. 1,ima Santos / Z(N) symmetric quantum spin models"

450

at the self-dual points X, = ~, i = 1. . . . . N. The relations (5.19) and the fact that the hamiltonian (5.5) is the first conserved charge permit us to obtain directly its ground state energy by the first derivative with respect to X L of the free energy of the classical self-dual Ashkin-Teller model. In fig. 3 we show schematically the equivalence between the eight-vertex and Ashkin-Teller models. From this figure, it is clear why for the first model the generator is the row-to-row transfer matrix while for the second one the generator is the diagonal-to-diagonal one. 5.2. Z(5)

The self-dual hamiltonian (3.4) for N = 5 is given by

-H=

E { Cl[ok + O[] + C2(0~ + Oa+2)).

(5.20)

k

The L - A pair equations are solved in appendix C. Contrary to the N = 4 case the ratio of couplings C2/C 1 is now fixed. This means that we can only find a L - A pair with the structure (4.5) for isolated points in the space of coupling constants. Apart from the 5-Potts point C2/C 1 = 1, we can also have the following two solutions: Ct

C2

-

C2

C1

sin ~rr sin ~" ' sin X~r 5 sin~Tr'

L a = 1 + X(p a + p£) +

Lk

1+

~(C2 +)~C1) C~ + XC2

(p~

-k p/2),

X(C2 + XCL) -1 CI+aC2 (0~+0~;)+a(0~+0/2).

(5.21)

(5.22)

The conserved charge can be calculated exactly as in the N = 4 case, by considering V~ = 02 + O~ 2. The first three local and non-local charges are again given by (5.8) and (5.10). In a similar way as for the Z(4) model (see appendix B) it is also possible to show that (5.11) is valid, i.e. all the motion integrals commute with each other. 5.3. Z(N), N > 5

The degree of complexity of eqs. (4.7) grows drastically with N. Solving eqs. (4.7) up to N = 9 we realise that an L - A pair may be calculated for the special couplings 1

C i - sin(Tri/U) '

i= 1,2 . . . . . N - 1,

and the general form for the operator L is given by

L k=l+

Y'~ I,,p~, I1= ]

(5.23a)

F.C.AIcaraz,A. LimaSantos / Z(N)svmnletricquantum,Vmlmodels

451

where

11=~,

, (ci++,ci)

lP=~i~=l= Ci+XCi+l ,

p=l,2

..... N-1.

(5.23b)

The structure for the A k operators is more complicated. However they can be expressed in terms of the lp, Cp; p = 1 . . . . . N - 1 given in (5.23) by inverting the relations (4.7). This can be done easily because the matrix Di/= l{i j} is cyclic, the result is

-1-1(,ko ke2 -k JJN1 E V'N 11 i2wmn/N

a,,j= NCi E

k=0 m=l

"-"n=0"ne

i=1,2 ..... N-l, a0,0 = 0,

a0,j = %, 0

j=0,1,2 ..... N-l,

j = 1,2 . . . . . N - 1.

(5.24a) (5.24b)

6. Exact solution for the Z ( N ) quantum models We will now find the ground state energy for the quantum hamiltonian (3.4) at the special point in the space of coupling constants

C,-

1

sin( ~rn/N )

,

n = 1,2 . . . . . N - 1.

(6.1)

In the last section we saw that at this coupling the hamiltonian has an infinite set of conserved laws, being the generator of charges given by

-oo (

~(~)=

17

k=+~

N-1

)

1+ ~ l,p] , n=l

]

(6.2)

where l, is given by (5.23b). By comparing (6.2) with the self-dual diagonal-todiagonal transfer matrix (2.19) of the classical model (2.3) we see that the above charge generator is exactly the transfer matrix at the couplings.

l,,= X(n)= Y(n)=exp{ U~=lflJxm[COS(2~mn)-- l]).

(6.3)

Consequently knowledge of the spectrum of the transfer matrix (2.19) at the couplings (6.3) enables one to calculate the spectrum of the hamiltonian (3.3) at the couplings (6.1) as well as the spectrum corresponding to the other higher charges, by using (5.7) and (5.9).

452

F.C. Alcaraz, A. l.ima Santos / Z(N) symmetric quantum spin models

To proceed further it is convenient to introduce the parameter a by writing

sin( a/2N ) )t = s i n ( r r / N - a/2N) "

(6.4)

Substituting (6.4) and (6.1) in (5.23b) produces

,, I

10=1,

sin(Trk/N+a/2N) 1,,= k1-I=0sin(~r(k+l)/N- a/ZU)"

(6.5)

Recently Fateev and Zamolodchikov [6], by solving the star-triangle relations for the self-dual classical Z ( N ) model (2.3), were able to find the free energy f per particle for a particular Z ( N ) model on the square lattice*. Their solution corresponds exactly to the Boltzmann weights (6.3), with /,, given by (6.5): :¢ sinh ~2ax sinh f (a) = - [%

~(~r - c0x sinh L2~rx(N- 1) dx

cosh2(

'7~x)co-G ~-TNx

x

(6.6)

Consequently (5.9) and (2.16) gives us the ground state energy for the self-dual hamiltonian

sinh ½~rxsinh ½~rx(N - 1) Eo=-Nfo

~osh-T(-½~xx~c-~sh_~ x

N-l dx-

1 sin(rrn/N)'

n=lE

(6.7)

while the ground state of the second local charge is

N1 i(2°)=D+½sinh -~ -

1

)

Eo- E sin(rrn/N) n

=

1

( ~")fo~¢ sinh ½~'x sinh ½rrx
- Nasinh 2 ~

(6.8a)

where D is a constant given by 2

D=

N 1

1 d (Zn=l/n) 2 d2~k 2

~()"

with similar expressions for the ground state of the higher charges. * The solutions for the triangular and hexagonal lattices were derived more recently [12].

(6.8b)

F.C Ah'araz, A. Lima Santos / Z(N) ~3,mmetric quantum spin models

453

7. Summary and conclusions

We have found an infinite set of conservation laws for self-dual Z(N) models at particular points in the parameter space. Starting with a general self-dual hamiltonian N

1

- H = E ¢p;', 11-- I

with C,, i = 1,2 . . . . . N - 1, real, we look for a Lax pair L - A satisfying (4.3) which guarantees the existence of the infinite set of conserved charges. We were able to find L - A pairs for three different families of hamiltonians: (i) The quantum N-Potts model (C i = 1, i = 1. . . . . N - 1). (ii) The Z(4) model with general couplings. This case corresponds to the quantum Ashkin-Teller model and may be transformed into the X Y Z quantum hamiltonian. (iii) The Z ( N ) model where

Ci -

sin(rri/N)

"

i= 1 .....

N - 1.

Infinite sets of local and non-local charges can be obtained from the generators L and I given by (5.7) and (5.9) respectively. For all these solutions we show that the non-local generator ~ is the diagonal-to-diagonal transfer matrix T D of a corresponding self-dual classical model in (1 + 1) dimensions. Consequently the infinite set of charges is also conserved for these classical models. The corresponding classical models for families (i) and (ii) are the N-Potts model and the Ashkin-Teller model respectively. Family (iii) is related to the Z ( N ) classical model with the Boltzmann weights ,,-i

~=1,

sin(~rk/N+a/2N)

X<',;')': kYI:o s i n ( ~

+ ~-/-AI ~ ~ - N

) '

whose free energy per particle was calculated exactly by Fateev and Zamolodchikov [6]. This enables us to derive the ground state energy for the family (iii) of hamiltonians. We believe that for N > 4 the family (iii) of hamiltonians corresponds to the bifurcation point of the classical Z(N) models where a massless phase appears [7]. We would like to thank M.N. Barber, M. Batchelor, B. Davies, G. Newsam, P. Pearce, S. Roberts and P. Rujan for relevant discussions and T. Lynam and A. Zalucki for help with the manuscript. This work was supported in part by the Australian Research Grant Scheme and by Fundaq~o de Amparo h Pesquisa do Estado de S~o Paulo, Brasil.

454

F.C. A h'araz, A. l,ima Santos / Z(N) symmetric quantum spin modeA'

Appendix A LAX PAIR FOR THE Z(4) MODEL

The aim of this appendix is to derive the L - A pair for the model Z(4). Using eqs. (4.10) and (4.11) we obtain the following relations A I , j P + A 2 , j Q = III~I,jCj,

j = 1,2,

(A.1)

At,jR + Az,jS=

j=l,2,

(A.2)

P = 2 1 2 - 1 2 - 1,

Q = I i ( l 2 - 1),

(A.3)

R = 21~(l 2 - 1),

S = 12- 1,

(A.4)

lzB2,jCj,

where

which permits us to obtain Sll[~I,.!

Al"i =

A 2, j =

The fact that

A1. 2 =

-

PS

QI2B2, j QR

Ci'

Pl2B2. s - Rllfll, s PS - QR

j = 1,2,

(A.5)

j = 1,2.

(A.6)

A2, x gives us the relation between l 2 and l I 12 = l I

C2 +

Cll z

(A.7)

C 1 + Czl 1

Eq. (4.12) together with (4.9) gives us

B1,1

l1 1 -

12

Cl=-B1,3=B3.3,

Bl,o=B1,2=B1,3=B2,3=O, (A.8)

and no constraints appear between the coupling constants C 1 and (72, which is a peculiarity of the Z(4) model. For N > 4 constraints between the coupling constant will appear (see appendix C). Summarizing, the L - A pair for the Z(4) system is given by: ) k ( C 2 -1- Cl~k )

L k = 1 + X(p~ + p~) +

C l + XC2

P2

(A.9)

and 3

3

Ak = E E a, ' ,OkOJ L, i~0 ]=0

(A.10a)

F.C. A Icaraz, A. Idma Santos / Z(N) symmetric quantum spin modeL~

455

where 2X(1 a l , o = a3,o : ao, l = ao,3 =

(1

212C1

+ 12)C 2

+12)2--4X

2

(A.10b)

1 --/2 '

4X 2 a2, 0 = a 0 ,

2=a2, 2=

_

(1

+ •2) 2 - -

a* = a* = 1,3 3,1 = al,1 = a3.3

a l , 2 = a3, 2 = a2,1 = a2, 3 =

(A.10c)

C2 , 4~k 2

l 2 - i~k2 ~(1 -- 12)

Cl

2X(1 +

12)

-

(1 +12) 2 (1 + 1 2 ) 2 - - 4 X

C2 ,

(A.10d)

2

C2,

(A.10e)

(1 + 1 2 ) 2 - 4 ) k 2

and where

x(G+ c,x) l2 -

(A.10f)

C 1 + XC2 Appendix

B

INVOLUTIVITY OF THE MOTION INTEGRALS

In this appendix we want to show the involutivity of the motion integrals I,, for the Z(4) model. It is sufficient to prove that for arbitrary parameters ~ and if:

=0,

(B.1)

where £~°(X) is given by (5.6) and (5.7). We can show this [3] by finding a two parameter L - A pair associated with the non-local hamiltonian L(/~) satisfying the equation [£,0(#), .LPk(X)] =Ak+l(2t,ti)Lk(2t)-Lk(X)Ak(X,IX).

(B.2)

Due to the fact that Lk(2t ) depends only on the operators at the site k and the structure of eq. (5.12) lead us to guess the following form for Ak(X,/~) k+l

Ak(X,

) =

(B.3a)

H n=+OO

where N-1 N-1

Bk(X,ff)= Z

E b, jo~O/~_i,

i = 0 1=0

bo,o=O.

(B.3b)

456

F.C. Alcaraz, A. l.ima Santos / Z(N) s3,mmetric quantum spin models

Substituting (B.3) in (B.2) we have the following equation:

[L,+,(tt), Lk(X)]

Lk(II)L,_,(#)

+ Lk.x(#)Lk(tt)[Lk_,(#)

= Bk+,(X,g)Lk_I(I~)Lk(X)

- Lk(X)Lk+,(g)Bk(X

,

L,(X)]

,Is).

(B.4)

After a lengthy calculation we find that the coefficients of Bk(X,/~) are given by 4~//'2~'(~'2 -- P'2) b°'2 = b2'° = 2p,~.(~k 2 -- 1)(1 + ~2) --/*2( l + )k2) 2 '

hi'° = b°'l =

(B.5a)

P'(~'2 - P'2) (1 4- ~.2)(2p,)k - )t 2 + 1) ~2 - I -k 4 ~ ( ~ 2 - 1) b2'°'

b ' " = b3 3--/'2'-

i/~X+(/'2 2 - X-1 1 )

4/~2 x

2/*/-t2}t 2

bl. 2 = b2,1 = b3, 2 = b2, 3

X2~

~2(1

-}- ~ - - i - 1 ) b2,o ,

(B.5b)

(B.5c)

-1- X2) 2 -- ~(1 q- ~ 2 ) ( X 2 - 1) 2

+

2/.t2(X22 - 1)

b°'2'

(B.5d) where

x(c2 + c,x) ~2

C~ + XC2

+ c,.) ,

/*2 -

C 1 + ~C 2

(B.5e)

Appendix C LAX PAIR FOR THE Z(5) MODEL

We derive here the L - A pair for the hamiltonian (5.20). Eqs. (4.10) and (4.11) give us

A a . , P + A 2 , y Q = l l f l , ,.~,

j = 1,2,

(C.1)

AI.jQ + A2. jR = IzI~2 jCj,

j=l,2,

(C.2)

where P = 2 1 2 - I e - 1,

R = 2122- I , - I.

Q = 2 1 f f 2 - I, - I 2,

(C.3) (C.4)

457

F.C. Ah'araz, A. Lima Santos / Z(N) symmetric quantum spin modelv T h e fact t h a t BL2 = B2, x p r o d u c e s the c o n s t r a i n t

(c.5)

[Q108,. ~ - p[2t~2.1]C 1 = [ Ql2fl2, 2 - R l l f l l , 2 ] C 2 . In a s i m i l a r w a y the eqs. (4.12) a n d (4.9) give us B i , j ( l 2 -- 1) + n 2 , j ( l 2 - ll) = l l ' ~ l , j q ,

j = 1,2,

(C.6)

B l , j ( l 2 - 11)

j = 1,2,

(c.7)

+

B2,j(I 1 -

1) =

12"~2,jq,

[ ( 1 2 - 11)11~1,1- ( 1 2 - 1)/2"Y2,1] C1 = [(/2-/1)12'Y2,2-

(ll-

1)11"/1,2] C2- (C.8)

C o n t r a r y to the case N = 4, the c o u p l i n g s C 1 a n d C 2 a r e n o t free b u t have to satisfy ( C . 5 ) a n d (C.8). A p a r t f r o m the 5 - P o t t s s o l u t i o n , t h a t c o r r e s p o n d s to C I / C 2 = 1 a n d It = ~ (see sect. 5) we find two o t h e r p o s s i b l e s o l u t i o n s : C1 C2

sin 5q7• -

C 2 -1- ~kC1 11 = ~

sin lrr '

l 2 = )~

'

sin ~Tr

C1 -

C2

Cl + )tC2

,

(C.9a)

C 2 + ~C 1 ,

17 =

sin }~r

-

~,,

lI

= ~

(C.9b)

C 1 + ~C 2

T h e c o e f f i c i e n t s A~,j a n d B~,.j in b o t h cases are given b y

Rll]~I'I - Ql2t82" AI'I =

P R - Q2

R/1/~l'2 - Ql2/~2"2 c

(C.10a)

c1,

AI'2 =

(72,

Ao, 1 = - 2A1,1l ~ - 2A~.212 ,

PR

Q2

2,

Pl2/~2, 2 - QI1/~I, 2 A2'2 =

P R - Q2

(C.10b)

(c.a0c)

Ao, 2 = - 2 A 2 , x l 1 - 2A2,212, while

<"=

Bl'2 =

(l 1 -

1)llYl.

1 -- (l 2 --

<'

(C.lOd)

( l 1 -- 1 ) l l g l . 2 -- ( l 2 -- 11)123,2, 2 ( l l - 1)(12 - 1) - ( l 2 - l l ) 2 '

(12 - 1)12"/2.2B2,2

ll)12"},2,1

(ll -- 1)(12--

(C.10e)

(12 - 11)11"/1,2 1) -- (12-- ll) 2

Bz,o

T h e o t h e r c o e f f i c i e n t s are o b t a i n e d f r o m these b y u s i n g (4.9).

B1 0

O.

(C.lOf)

458

F.C. Alearaz, A. Lima Santos / Z(N) symmetric quantum spin models

References [1] R.J. Baxter, in Exactly solved models in statistical mechanics (Academic Press, 1982), and reference: therein [2] M. Lilscher, Nucl. Phys. BI17 (1976) 475; E. Baxouch, Studies in applied mathematics (1984) p. 151 [3] Y.A. Bashilov and S.V. Pokrovsky, Comm. Math. Phys. 76 (1980) 129 [4] R.B. Potts, Proc. Camb. Phil. Soc. 48 (1952) 106 [5] J. Ashkin and E. Teller, Phys. Rev. 64 (1943) 178; M. Kohmoto, M. den Nijs and L.P. Kadanoff, Phys. Rev. B24 (1981) 5229 [6] V.A. Fateev and A.B. Zamolodchikov, Phys. Lett. 92A (1982) 37 [7] F.C. Alcaraz and R. K~Sberle, J. Phys. A13 (1980) L153; A14 (1981) 1169; J.L. Cardy, J. Phys. A13 (1980) 1507 [8] G. von Gehlen and V. Rittenberg, Nucl. Phys. B257 [F514] (1985) 351 [9] C.J. Hamer, J. Phys. A14 (1981) 2981 [10] M. Kohmoto, M. den Nijs and L.P. Kadanoff, ref. [5]; V. L[bero and J.R. Drugowich de Fel~cio, J. Phys. A16 (1983) L413 [11] F,J. Wegner, J. Phys. C5 (1972) El31 [12] N. Onody and V. Kurak, J. Phys. A17 (1984) L615