Solvable nineteen-vertex models and quantum spin chains of spin one

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Solvable nineteen-vertex models and quantum spin chains of spin one

Makoto Idzumi, Tetsuji Tokihiro, Masao Arai

To cite this version:

Makoto Idzumi, Tetsuji Tokihiro, Masao Arai. Solvable nineteen-vertex models and quantum spin chains of spin one. Journal de Physique I, EDP Sciences, 1994, 4 (8), pp.1151-1159.

�10.1051/jp1:1994245�. �jpa-00246975�

J. Phys. I IYance 4

(1994)

llsl-l159 AuGusT 1994, PAGE llsl

Classification Physics Abstracts 64.60C 75.10J -02.90

Solvable nineteen-vertex mortels and quantum spin chains of spin

_one

Makoto Idzumi

(*), Tetsuji

Tokihiro and Masao Arai

Department of Applied Physics, University of Tokyo, Hongo, Bunkyo, Tokyo, i13, Japan

(Received

10 January1994, received in final form 12 April 1994, accepted 15 April

1994)

Abstract. We solved the Yang-Baxter equation for the R-matrices of three-state vertex mortels with ice condition, and obtained _a complete list of solvable mneteen-vertex mortels and

associated quantum spin Hamiltonians of spm one.

In this brief paper we report on a

complete

list of solvable nineteen-vertex models which

are classical statistical mechanics models _on square lattices

(see Fig.

l, which shows

visually

our

assumption

_on the Boltzmann

weights).

The _term solvable _means that the Boltzmann

weights

^of the model _or the R-matrices

R(u),

with

spectral

_{parameter tt,}

satisfy

the

Yang-

Baxter equation

(YBE)

_or the

star-triangle

^relation.

(R(u))u~c

defines _a

family

of solvable models. When the YBE is

satisfied,

the transfer matrices of the models in the

family

commute:

[T(u), T(u)]

= 0. It _means coefficients

T(")

_{of tt"} in

T(u)

form _{a set} of

commuting

operators.

It is believed that the existence of such

commuting

operators ^is a sufficient condition for

solvability

of the model [1, 2]. ^In

fact,

_{in many cases,} there exist methods to obtain _{an exact}

expression ^{for the free} energy: for

example,

_{one can}

apply

the Bethe Ansatz _or the quantum

inverse scattering ^method

[Ii

_{or some} other methods [2]. To _our

knowledge, however,

_we do

not have any

almighty

method to get the free _{energy even}

though

the R-matrix satisfies the

YBE;

_we must try to

apply

the Bethe Ansatz method _or the other _case by case

(but

how to

solve the models is non the

subject

of this

paper).

We also

give

_a list of the

integrable

quantum spin chains of

spin

_one, which _are related to the above vertex models

through

the transfer matrices. The Hamiltonian _is defined

by

a

logarithmic

derivative of

T(u).

The _set of

mutually commuting

operators

(T("))n=o,1,2,...

includes the

spin

Hamiltonian. We have said the _spin chain is

integrable

because of the existence of such _many

commuting

operators.

This _paper is

organized

as follows. First _we define vertex models and associated quantum

spin

chains in _a

general setting,

and

then,

_we shall define the nineteen vertex model and

associated spin chain of _{spin one} which _we shall consider in this _paper. After

that,

_we descnbe

(*) Present address: Department of Mathematics, ^Shimane University, Matsue, Shimane 690, Japan.

II52 JOURNAL DE PHYSIQUE I N°8

tl"~> ~ à=~,

,

~ C"~> ~

g~ ~ ~ ~,

P"~,

, ,

~~~

o=~.

Fig. l. Allowed nineteen _vertex configurations.

Dur method _ta find Dut ail the solvable models. We note that _we made fuit _use of Mathematica

m the _course of calculations. Dur strategy was

e~hausting

ail

possible

_cases with the

help of

the

computer. Results _are summarized in three tables:

complete

lists of solvable nineteen-verte~

models and associated solvable quantum spin Hamiltonians

of

spin _{one are} shown in tables I and

II;

_we aise show related solvable t- J models _m table III. There _we shall

point

Dut that _some

mortels in _Dur lists _were

already

known. For the methods of solution for the known models, we

shall refer _ta the

original

_papers. As _was mentioned above _we shall _net be interested in the method of solution.

A vertex model _{on a} two-dimensional _square lattice is defined _as follows. Let _a spin ^variable live _on each bond of the lattice and take values in _some range, say

I;

for

example

I

=

(1,

0,

-1).

TO each vertex

(or site)

attach _a Boltzmann

weight

P

j~~OE ^{~ "}

vp

~

oe

N°8 NINETEEN-VERTEX MODELS AND SPIN CHAINS OF SPIN ONE l153

where _{~t, v, a,}

fl

E I. The

partition

^function is

z

~ ~

_j~~OE

UP configurations ^vertices

# tr

T~,

where the T is the transfer matrix

j'OEl"'CEN j~~loElj~~20E2 j~~NOEN

ôl" ôN

~

~261 ~362 ~lôN

~l>... ~N

We assumed that the _size of the lattice is M _x N and that boundaries _are connected ta the other sides sa that the lattice is _{on a} torus. In the

thermodynamic brait,

_{we get}

Z~Af

where Ao is the _maximum

eigenvalue

of the transfer matrix

(we

_assume here that it is net

degenerate).

One

goal

of statistical mechanics is _to find _{an exact} expression for the

Ao.

Below,

_{we suppose} that the Boltzmann

weights

_are

parametrized by

_some functions of _a variable _{u E}

C,

called

spectral

parameter, _as R(~1).

A quantum

spin

chain is associated with the vertex model. The Hamiltonian _is defined

by

H ₌

lT111)~ £T1~l)1)1...)1 l~-~.

It is related _to the Boltzmann

weights

^{of the} vertex model

through

the transfer matrix. We note that the Hamiltonian thus defined

has,

in

general,

_{a very}

complicated

form and _can contain

long

_range interactions.

We _{now assume} that the Boltzmann

weights

_or the R-matrices

satisfy

the

(additive) YBE(~

R(~1)j(j(R(~1+ v)((j(R(v)((((

₌

R(v)j(j(R(~1

+

v)j(((R(~1)((((. Il)

From the YBE _we get

Riv Îl)iTiv)

_@

Till))

=

iTill)

_@

Tiv))Riv Îl)

12)

where the _{T is} the

monodromy

matrix defined

by

~~~~~Î~~"

ÎÎ ~ _~~~~~~~l

_~~~~~~~~

_~~~~~ÎÎ~~

~2 _;. ~"

Tàking

traces of the bath sides _we have

viii)> Tiv)1

= °. 13)

(~ ^{It is known that there} exist nonadditive YBES which produce solvable models. An example _is the chiral Potts model [4, Si. ^{In this} paper we shall net be concerned with such models related to

nonadditive YBES.

l154 JOURNAL DE PHYSIQUE I N°8

Table I. Vertex

weigllts

^{of solvable 19} vertex mortels.

Here,

U _e e", ik ₊

+1,

^â~ +à -1

= 0,

fl~

(à 1)fl (à

_1)~

= 0.

vertex

weight #1

#2

#3 #4 #5 #6

a

e~~

_{ch~l +}

chqsh~l

ch~l+

chqsh~l

+ ii~1 1

~ ~ ~

(ô 1)(1

eV) ^-eV + e~U

~~~

l +

(à

₊

fl

1)eU ^{2 eV}

e~U = a = a 1 ₊ b b ₊ eV

1

g 0 0 0 0 +b +b

o

e°~

= a ch~l

chqsh~l

+13~1 1 ₊ b b ₊ e~

p 0

sh~sh~l shqsh~l

_14~1 0 0

#7 #8 #9 #10

1 ~~1 ₊ ~1~

uj4

^u4 ~ sh(~1 +

q)sh(~1

+

2q)

~ ~ ~

shqsh2q

[11

+ ~12)

~

[iii u4)

^{~~ Sh~lShl~1}

ⁿ⁾

~

u3ji

^{3 +}

V~~i4) ^ShnSh2q

2

1

-U(2+U~)

_b+

3 U3

1 _~1

j(4

^U~

~~~)~

^~~

s q

i1~l +

~U~(1 _{U~) +(}

^~

~)~/~U

_{b ii}

)

3 2 _s q

1 1

~ ~~ ^{l +}

VÎ

~

2~

₊

( ^~~~)

sh~lsh(~1+ q)

2~

^~ ₂^~ ₂

3 ₊

VÎ

~

~

shqsh2q

U(-1

+ U

Sh~lsh(~1 +

q)

%2

§ (~~l

^{+ ~l~ ° ~ ~~}

ShqSh2q

N°8 NINETEEN-VERTEX MODELS AND SPIN CHAINS OF SPIN ONE l155

Table II. Solvable

spin-1

quantum

spin

Hamiltonians. ik,

à, fi

_{are as} above.

coefficients

#1 #2 #3 #4 #5

a O

chq

+

shq -chq

+

shq

-212 + 13 + 14

~ à ₊

fl

fl IA C)

₊ 30

3chq -3chq (ii

712 + 613) o

1 O

chq -chq -12

_{+ 13} 0

à

IA

+

C)

₊ 20

3chq -chq iii

512 + 413 0

e O

chq

+

sh~ -chq

+

shq

-212 _{+ 13} + 14

Il

+

1)

a

q -O

-chq chq

212 _{13 -a}

~t -O

-chq chq

_{12 13} 0

#6 #7 #8 #9 #10

~ ~~ ~ ~~~ ~ ~

~~ Î121

~

-2 1 -21 ₊

8VÎ

^~

~~~~

2 _s

2q

~ ~

~~ s~q

~ ~~ ~

~~

^~

s/ÎÎ~~

~l

-j(3+2ii+12) ~~~~ ^-9+4v$+4( _~~~)~/~ a-ii~)(~

S n

~ ~

~~

shlq

~ ~

~~ Î~Î

l156 JOURNAL DE PHYSIQUE I N°8

Table III. Solvable t J mortels. Here,

A,

C,

O,

_{q are}

arbitrary

constants, and ik ₌ +1.

#1-#4 correspond

to tllose in tables I and II.

#1 #2 #3 #4

t 0

-shq -shq

-14

Jz

2(A C)

0 0 2ii

Ji 0 0 0 212

V

IA

+

C)

₊ O

2chq

0 ii

+13

Expanding T(u)

₌

£ T(")~1",

_we get

[T(~), T(")]

= 0. It _means

(T("))

forms _{a set} of commut-

ing operators, ^The existence of such _a set is

regarded

_as a suilicient condition for

solvability.

In many cases, one can find _an

expression

for the free _energy

by

_means of _some appropriate methods:

equation (2)

is _a

starting point

of the quantum inverse

scattering

method (1] ^and equation

(3)

is _a basic relation of Baxter's

commuting

transfer matrices method [2].

Again

_we

remark that _an

almighty

method of solution is not

known;

in

general

_we must try to

apply

the Bethe Ansatz method _or the other _case

by

_case.

Let us

specify

the model which _we shall consider in this _paper. The nineteen-verte~ model in this _paper has the range of spin variable

(1,

0,

-1)

and allowed vertex

configrations

_as shown in

figure

^1. The other

configurations

_{give zero} Boltzmann

weights.

For each horizontal

edge,

we

assigned

_a

right-arrow

for spin variable +1, a left _arrow for -1, and

nothing

for 0,

respectively.

For each vertical

edge,

_we

assigned

_{an up-arrow} for +1, _a down _arrow for

-1,

and

nothing

for

0, respectively.

We have assumed the

following symmetries: Ii)

the ice r~lle

(or

the ice

condition)

RS)

= o uniess ~t + a = v +

p;

(ii)

the usual

spin up-down

symmetry

j~~OE j~~~>~OE,

HP -V,-fl'

(iii)

_an invariance with respect to reflections

~~OE ~OE~

~Pv

vp pv a~.

In this _{sense, it} is _a direct extension of the well-known _six-vertex model (3, 2]. ^Trie q~lant~lm

spin

Hamiitonian

of

spin _one is defined _as before

through

the transfer matrix of the nineteen-vertex model. The ice rule results _{m a} relation

l~

~Zj ^~

, ,

1-e-, the Hamiltonian commutes with the total

projected spin

S~

=

£~ Sj.

In the

following

_we

further make _a standard assumption

li(Ù)

_{Î~, l~~p}

_ô~flôv~.

N°8 NINETEEN-VERTEX MODELS AND SPIN CHAINS OF SPIN ONE l157

Then the Hamiltonian will contain

only nearest-neighbor

interactions

~ "

i

~~~~>

~°J-i~J-i llRi~l)iiiiil«-~ ôai+iôi+i

ôoe~~~.

We have said that this quantum

spin

chain has

spin

_one because to each site _a three-dimensional vector space is attached. Later we will _see that this is

actually

written in terms of the

spin-one representation

^of

su(2)

generators.

We have solved the YBE

Il)

for the R(~1)

given

in

figure

1 and found ail solutions. Dur method to find ail R-matrices

satisfying

the YBE is quite naive and

straightforward.

We _assume that

Ii)

the R-matrix has

only

19 _nonzero entries with the

symmetries

described in the last

paragraph (see Fig. l); (ii)

the R-matrix R(~1) ^is an

analytic

function of _{~1 m a}

neighborhood

of _~1= 0;

(iii) Rio)

=

P;

and

(iv)

it satisfies the additive YBE

il).

TO find _a

possible

form of R(~1) ^let us

expand

it in _~las

Rj~)

=

R(o)

_~

~j~(1) _~ ~2 j~(2) _~

where, by assumption, R(°)

= P.

Substituting

this

expansion

into the

YBE,

_{we get} several relations to be satisfied

by

the coefficient _matrices R(~) _We wish to salve these

equations

and to find _a solution

R(~),

i

= 1, 2,.

Luckily

_we found that it _was

possible.

In

fact,

in the _course of

calculations,

_we found that

la)

the coefficient matrices

R(~),R(~),.

_{were ail} written in

terms of entries of the

R(~)

_{and 16) equations for the entries of the}

R(~)

_{could be solved: there}

were a finite number of

possible

forms of

R(~)

_and

me e~ha~lsted all

possibilities.

In this way we

got ^ail

possible

solutions

R(~).

The observed fact

la)

_is known _as the

tangential star-triangle hypothesis

_[6],

which,

to _our

knowledge,

has net been

proved ngorously (but

is believed to be

true).

We observed that this

hypothesis really

holds for the R-matrices _we have assumed.

Dur results _are summarized in tables I-III. In table I _we show the

complete

list of solvable nineteen-vertex models

(three-state

vertex

models)

with the

prescribed symmetries; they

_are

classified into ten groups for convemence. Table II gives the

corresponding

quantum spin Hamiltonians of _{spm one;} this table shows coefficients of the Hamiltonians, which _are defined

by

N

~

~~J,J+1>

j=1

Hj,j+i =a(S)S)~i

+

S)S)~~)

⁺

fl(Sjsj~i)

+

i(S)S)~i

+

S)S)~~)~

+

ôls]S]+1)~

+

fllsj Sj+1)~ [Sjsj+i

+

S]S]+1)~ lS]S]+i)~l

+ q

l§,j+1

+

~l(Sj

+

Sj~i)~, Pj,j+i =(Sj Sj+1)~

+

(Sj Sj+i)

-1,

where S~, _{= x,g,z, are} the standard spin operators of spm one

(the

spin-one representation of

su(2) generators)

10

0 0 -1 0 1 0 0

~~

V~

~~

~~

~ÎÎ ~~

~~

ÎÎ ^l~

Let _us point eut known and solved models _{m our} tables. First _we remark that model

#1

is

trivially

solvable _as there _{is no} interaction. The Hamiltoman #10 in table II is the _one

l158 JOURNAL DE PHYSIQUE I N°8

obtained

by

Zamolodchikov and Fateev (7]. ^This is a direct

land non-trivial)

extension of the well-known XXZ Harniltonian

Ian anisotropic Heisenberg spin Hamiltonian)

to the _case of spin _one. For _a method of solution _see their

original

_paper. We note that model

#7

_can be

obtained _{as a}

special

_case of

#10

in the rational limit. Sutherland [8] solved the model

N

H = q

£ _(,j+i

_n

" +1,

j=i N

= q

£((Sj _Sj+1)~

₊

(Sj Sj+i) I),

j=1

which is included in #4 in table

II;

_a solution of this model

by

the quantum ^inverse

scattering

method is described in reference [9].

Takhtajan [loi

and

Babujian iii]

solved _a

spin

chain defined

by

~

N

~

i~

^{~~~ ~~~~ ~}

~J,J+1 21),

=

£(-Sj _Sj+i

₊

_{(Sj Sj+1)~ 31),}

2

~

J=1

which is induded in

#7,

and

Klümper

_[12] solved

N

H _{= a}

£(-(Sj _Sj+1)~

₊

_I),

j=1

which is induded in

#5.

Model

#6

_was solved in

Klümper

_[13]. For the methods of solutions

we refer _ta the

original

_papers.

It _seems that _Dur models #2, #3, #8, and #9 are new

(We

wish ta thank the referee for

this

information).

Dur list of

spin

Hamiltonians in table II contains solvable t J models _as

special

_cases. The t J model is _a model of

tight-binding

electrons with

correlation,

and defined

by

the so-called t J Hamiltonian

Ht-j ₌ t

£ Pc$cj~P

₊

_h-c-)

<1,J>

+

~j _(Jz (si sj)

+ Ji

(sis)

+

s$s))

+

Vninj )

<i,J>

where

ni ⁺

~j c$cw

= 0 _or 1 def. of _an action of

projection P),

«

8~ %

~ -C$OE$~icia"

~ ~i

Table III

gives

coefficients of the solvable _t J Hamiltonians.

Among

them the model #4 is

interesting:

^{for both} t ₌ +1, there _are solvable _cases

(Jz

=

Ji,V)

₌

(2, -1/2), (-2,1/2),

N°8 NINETEEN-VERTEX MODELS AND SPIN CHAINS OF SPIN ONE l159

(2,3/2), (-2, -3/2);

which _are

isotropic

t J models solved

by

Schlottmann [14]. ^We note that there _are other solvable _cases: for both _t

= +1, these _are

(Jz, Ji,V)

=

(2,

-2,

-1/2), (-2, 2,1/2), (2, -2,3/2), (-2,

2,

-3/2);

these _are

anisotropic

t J models. We do not know

whether these

anisotropic

models have

already

been solved _{or not.}

Finally

_we remark _on the

symmetries

_we have

imposed

_on the R-matrices. Dur interest _was

mainly

in _a natural extension of the six-vertex model and the XXZ model to

higher spin

_cases from the

view-points

of condensed matter

physics.

We

thought

that _our choice of

symmetries

were natural for this _purpose. If _{we were} interested in _{a more}

general

class of solvable

models,

we had to relax _some conditions _on

R;

_{one can} find _an

investigation

m this direction in Perk and Schultz

[15],

where _some models _not

presented

in the present _{paper were} obtained.

References

[1] ^{Kulish P-P- and} Sklyanin E-K-, Lecture Notes _m Pllysics 151

(1982)

fil.

[2] ^Baxter R-J-, Exactly solved models _in statistical mechamcs

(Academic

Press, London,

1982).

[3] ^Lieb E-H-, Pllys. Rev. 162

(1967)

162; Pllys. Rev. Lent. 18

(1967)

1046, 19

(1967)

108.

[4] Au-Yang H., Mccoy B-M-, Perk J-H-H-, Tang S. and Yan M., Pllys. Lent. A 123

(1987)

219.

[5] Mccoy B-M-, Perk J-H-H-, Tang S. and Sah C.H., Pllys. Lent. A 125

(1987)

9.

[fil Jimbo M. and Miwa T., Nucl. Pllys. 8257

(1985)

1.

[7] Zamolodchikov A.B. and Fateev V.A., Sov. J. Nucl. Pllys. 32

(1980)

2.

[8] Sutherland B., Pllys. Rev. B 12

(1975)

3795.

[9] ^{Kuhsh P-P- and} Reshetikhin N.Yu, Sov. Phys. JETA 53 (1981) 108.

[loi

Takhtajan L.A., Pllys. Lent. A 87

(1982)

479.

[Il]

Babujian H-M-, Pllys. Lent. A 90

(1982)

479; ^Nucl. Pllys.

8215[FS7] (1983)

317.

[12] Kliimper A., Europllys. Lent. 9

(1989)

815.

[13] Klümper A., J. Pllys. A: Math. Gen. 23

(1990)

809.

[14] Schlottmann P., Pllys. Rev. B 36

(1987)

5177.

[15] ^{Perk J-H-H- and} Schultz C.L., Proceedings of RIMS Symposium 1981, Jimbo and Miwa Eds.

(World

Scientific Publishing,

1983).