Solvable weak-graph duals of partially frozen vertex models

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Solvable weak-graph duals of partially frozen vertex models

M. Kolesík, L. Šamaj, P. Markoš

To cite this version:

M. Kolesík, L. Šamaj, P. Markoš. Solvable weak-graph duals of partially frozen vertex models. Journal

de Physique I, EDP Sciences, 1992, 2 (7), pp.1317-1323. �10.1051/jp1:1992212�. �jpa-00246623�

Classification Physics Abstracts

05.50 75.10H 64.60

Solvable weak-graph duals of partially fkozen _vertex models

M.

Kolesik,

L.

lamaj

_{and P. Marko§}

Institute of Physics, Slovak Academy of Sciences, D6bravski cesta 9, 842 28 Bratislava, Czechoslovakia

(Received

6 December1991, revised 16 March 1992, accepted 27 March

1992)

Abstract. A family of q-state vertex models _{on a square} lattice is solved exactly using the

generalized weak-graph transformation. Concrete physical symmetries of vertex weights turn out to be closely related _to abelian _groups. The possibility ofoccurrence ofthe first-order phase

transition is discussed.

1 Introduction.

A most fruitful

approach

to construct and solve the

integrable

models is based _on the

Yang-

Baxter

equation ii, 2],

which guarantees the existence of

commuting

families of the transfer matrices

parameterized

by _a

spectral

parameter. ^In certain

subspaces

of parameters of inte-

grable

models

(e.g.

_on surfaces _{a + c}

= + d _{or a +} d

= b_{+ c} of the 8-vertex model without _a field _on the _square

lattice)

_a trivialization _occurs: the

eigenvector corresponding

to the

largest eigenvalue

is _common for the whole set of transfer matrices which thus commute on this rel- evant one-dimensional _space [3]. ^{For such so-called} "disorder

solutions",

the free _energy is _an

analytic

function of model parameters and the correlations exhibit

a lower-dimensional be- haviour. The disorder solutions have their

analogues

in

spin

systems

(for general

methods and references _see Ref.

[4]).

In this _paper,

we present a

family

of

exactly

^solvable q-state vertex models

(q-SVMS)

_on

the _square lattice which _are

closely

related to the kind of trivialization observed for disorder solutions. The cornerstone of _our

approach

consists in the _use of _a

generalized weak-graph

transformation

(GWGT)

[5, 6]

mapping

the

original

model with _a

specific

symmetry of vertex

weights along

_one lattice direction onto the

partially

frozen _one, which therefore becomes

effectively

one-dimensional. We formulate the mathematical criterion for

constructing

the models and find its relation to the classification of abelian _groups of _a

given

^order _q. ^This

fact enables _us to look for

all,

from the

point

of view of the

weight

symmetry

non-equivalent,

solvable

q-SVMS

in _a

unique

_way.

According

to the structure of the

corresponding

_group, the symmetry ^{of the considered} models _appears to be related to the

Z(n) (n

<

q) symmetries,

^and

JOURNAL DE PHYS>QUEI T 2, N' 7, JULY >992 48

1318 JOURNAL DE PHYSIQUE I N°7

resembles the _one of the chiral Potts models initiated in reference [7]. ^In

spite

of the

relatively simple thermodynamics

these models _{are of} interest _{as a} contribution to the

understanding

of the

phase

structure of

general

vertex models _on the _square lattice.

The paper is outlined _as follows. In section 2, _a suitable

weight-matrix

formalism is

presented

for the

q-SVM

with certain symmetry

properties along

_one lattice

direction,

and the method of

searching

for _our set of

exactly

solvable

q-SVMS

is described. The connection with the group

theory

is

given

in section 3. The critical

properties

and the

possibility

of _occurrence of the first-order

phase

transition _are discussed in section 4.

2. The method.

We consider _a

q-SVM

_{on a square} lattice of M _rows and N columns with toroidal

boundary

conditions. Each lattice

edge

_can be in _one of _q distinct states s = 1,

2,

, q. With each lattice site

(m, n)

_we associate _a vertex

weight

_w~~s~s~s~ which

depends

_on the states _si,s2> s3> s4 of

incident

edges.

The

equilibrium

statistics of the

q-SVM

is determined

by

the

(dimensionless)

free energy per

site, f((w)),

defined _as

-f(lwl)

=

~jrf~ AID Z(lwl) (la)

where

~~~~~~ ~ ~

~SiS2S3S4

(~~)

(S) (m,11)

denotes the

partition

function with the summation

going

_over all

configurations ofedge

states and the

product being

_over ^{all vertices.}

In

general,

there exist q4 different

configurations

of

edge

states around _a vertex. We order q4

temperature-dependent weight

parameters into the matrix

~i~ll ~i~12 ~i~lq

~i~21 ~i~22

W2q

=( j, ⁽²⁾

~ql

_{~Vq2 ~i~qq}

where

W;j

^{denotes the} _q x q matrix of q~ vertex

weights

^which

correspond

to the _{same con-}

figuration

^of

edge

states

along

^{the vertical line}

(column)

of the vertex, si "

I,

_{s3 "} j. ^The columns

(rows)

in

W;j

are also indexed from left to

right (from

_up to

down)

as 1,

2,..

,q; the

element

(Wij)ki corresponds

to the _vertex

configuration

of

edge

states _{s2 "}

k,

_s4

"

along

_a

rOW,

(W;j)kl

_{" W;kjl.}

The above form of W _can

suitably

reflect the symmetry

properties

of vertex

weights along

one

direction,

in

particular along

the columns. If the

q-SVM

has _no symmetry then the matrix W is

represented

_as the _sum of direct products

q

W ₌

~j _E;j

_©

_{W;j. (3)}

I,I=i

Here, the _{q x q} matrix

Eij

has

only

_{one non-zero} element

(E;j)ki

_"

6;k6ji

and

W;j #

Wki for

(ij) # (ki).

Once _a symmetry of vertex

weights along

columns is present, W _can be written _as

W=~jC;©li~

p

₍₄₎

;=1

where _{p <} q~, Jiq

# Rj

for I

# j

and Cl

C2, _Cp

_can be

expressed

_as the

disjunctive

_sums of matrices

E;j.

^Thus no two distinct matrices Ci _{possess nonzero}

(unity)

elements _on the _same

place.

Each GWGT of vertex

weights,

which leaves the statistical _sum

(16) invariant,

_can be

expressed

as the

similarity

transformation that acts

separately

_on the column and _row

edge

states:

W =

Q

W

Q~~

,

(5a)

Q=U4lV. (5b)

Here,

U and V _{are any q x q} invertible matrices. In what follows _we let V to be the

identity

matrix.

We _now

explain

the strategy ^of

constructing exactly

solvable

q-SVMS

which symmetry con-

sists in commutative

properties

of the matrix set

(Cl,

_C2>

,

Cp)

defined in relation

(4).

Let all matrices

C;

_can be

diagonalized by applying

the _same

similarity

transformation

(5a,b) (I,e.

all matrices C; commute with each other and their _common

eigenvectors

constitute the _rows of

U). Then,

the transformed matrix W takes the

block-diagonal

form

q P

W ₌

~j

_fBwii

,

W;; ₌

~j _~j;Rj

;

(6a, b)

i=i I=1

ii;

is the ith

eigenvalue

of

Cj.

To find the free energy per

site,

_we need _an additional informa- tion about the structure of matrices W;;. if there exists _one

matrix,

_say

Wii,

whose

positive

elements _are

larger than,

at least

equal

to, the

corresponding

elements

(in

the absolute

value)

of

remaining

matrices

W;;,

then

-f(lwl)

₌ In Amax

(7)

with Amax

being

the

largest eigenvalue

of Wii To prove our claim _we consider the row-to-row transfer matrix of the transformed system

which, owing

to the

explicit

form of W

(see

relation

(6a)),

is

diagonal.

Each of its q/~

diagonal

elements is

equal

to the trace _{over a}

product

of N matrices chosen from the set

(Wii, W22,

,

Wqq).

^{Provided that} for _an

arbitrary

vertex

configuration

of two

edge

states s2 "

k,

_s4 "1

along

_{a row} it holds

(ifii)~i

₂

j(fil~~)~ij,.

,

j(fil~~)~ij, (8)

the

largest eigenvalue

of the _row-tc-row transfer matrix is

equal

to

Tr(W()

which

immediately

leads to

(7).

Note

that,

in the

thermodynamic limit,

each column

edge

is in state I under the above conditions and _so the system is frozen

along

the direction of the

supposed

symmetry.

3.

Exactly

solvable

q.SVMS.

Let _us

study

the

q-SVM

for which all

possible

vertex

configurations along

columns _are

allowed,

I-e- the elements of matrices

Cl,C2, .,Cp

_cover

disjunctively

_everyone of q~ matrix places.

Our task then consists in

searching

for all matrix sets

(Ci>C2; ,Cp)

with the

following properties:

(I)

^matrices

^Ci

commute with each

other;

(it)

^{the element}

(C;)jk (j,

^k ₌ 1,

2,

,

q)

^is non-zero,

equal

to

unit,

for

just

_one I

= 1,

2,

, or p.

1320 JOURNAL DE PHYSIQUE I N°7

Table _{I. Number}

ofnon-isomorphic

abelian _groups with _q elements.

q 12 3 4 5 6 7 8 910

number of I I 1 2 3 2 1

abelian groups

With

regard

to these

requirements, only

_{p = q}

permutation

matrices _can constitute the

required

matrix set. The

property

^of

commutability (I)

also

implies

that these

permutation

matrices form _{a group.}

The _group

theory provides

the numbers of

non-isomorphic

abelian _groups of order _q

(see

e-g-

[8]).

^For _{q <} ¹⁰ we present them in table I.

For each _q, there exists _one

cyclic

_group

represented

in _{our case}

by

matrices

0 0 0 0 0 0 0 0

Cl"

^~

>C2"

^{~ ~}

;

>Cq=

^{~ ~ ~}

(9)

0

0

l

0 0

0

0

In view of

representation (2)

and

(4),

such _a choice of

commuting

matrices

corresponds

to the

physical

model whose vertex

weights depend

_on the relative difference between the two

states of column

edges,

_s3

si(mod q).

This

Z(q)

symmetry of vertex

weights along

^columns

is

represented graphically

for _q

= 4 in

Fig.

la:

solid circles denote the four

possible

values of _si and of _s31

a vertex

configuration

of

edge

states

along

the column

(si> s3)

is

represented by

_an oriented line

connecting

^the

respective

circles

(an

_arrow goes-out ^the state _si and

goes-in

the state

s3)I

one

diagram

contains all

configurations (si> s3)

which _are characterized

by

the _same

weight

matrix J~i.

The kth

(k

₌ 1, 2,..

,

q) eigenvalue

of

Cj

is

given by

"

'~

~~~ ~~~ ^~~~~ ~~' '~~~~ _"

~XP(2~ilq).

~~

The

diagonal

blocks of W

(see (6b))

thus read

q

W;;

₌

~j

cy(~~~)(3~~)Rj

(ll)

I=1

q

Clearly,

for _an

arbitrary

choice of

positive Rj

the value of each element of matrix

WI

i ⁼

£ _Rj

I=i is

larger

^than

(equal to)

the absolute value of the

corresponding

elements of _any W;;

(I

₌

2,.

.,

q).

^{That is}

why

the

largest eigenvalue

of

Wii

Amax determines the free energy per site

through

relation

(7).

As is shown in table I, for certain values of _q there exist, besides the cyclic _group, also other abelian _groups. The last

correspond

to

particular

direct

products

of

cyclic

_groups of _a lower

4

Q C

~~

i

Q

i

Q

~

~i ^Ri

^{R~ R~}

b)

j _j~

i

C

Fig. I. The graphical representation of the vertex weights _symmetry

along

columns for

a) Z(4)-

symmetric model;

b)

the model induced by the direct product of two cyclic _groups of order 2. For explanation see the text.

order which _are not

isomorphic

with the

cyclic

_group of the

given

order _q. For _q

= 4, for

example,

_we have the commutative set constructed from the direct

products

of 2 _x 2

cyclic

matrices:

~

Ii _oj ji _oj

~

_ji _{oj jo ij}

~~ ° i ~

° i ^{' ~~} ° i ~

i ° ^'

~~~~i °~~~° il' _{~~~~i °~~~i °~'}

~~~~

The symmetry of the consequent

physical

model differs from that of the

Z(4)

model

(see Fig.

lb). Considering

the

explicit

forms of

diagonal

blocks

W;;, f((w))

is

again yielded by

^the

4

largest eigenvalue

of the ~'dominant" matrix _sum

~jRj.

_For

q = 8, there exist further two I=i

abelian _groups,

non-isomorphic

with the

cyclic

_one, constructed from the direct

products

of three 2 _x 2 _or two~ ⁴ x 4 and 2 _x 2~

cyclic permutation matrices,

and _{so on.} The role of the

q

"dominant" matrix is

always played by

the _sum

~j _Rj.

I=i

Every

_group

presented

in table I represents _a

large

class of

equivalent

abelian matrix sets connected

by

_a

similarity

transformation. The

isomorphic

abelian matrix sets of the

given

class lead _to

physical

models which have the _same

topology (more precisely, symmetry)

of vertex

weights

and _can differ from each other

only by relabelling

of

edge

states. For

instance,

the q = 6

cyclic

_group and the _group of direct

products

of 2 _x 2 and 3 _x 3

cyclic permutation

matrices fall within this category ^of

equivalent

sets.

1322 JOURNAL DE PHYSIQUE I N°7

4. Critical

properties.

In

practice,

_our treatment reduces the twc-dimensional

problem

to the

problem

of _{one row} intersected

by

frozen columns. Since each _non-zero element of the site-to-site transfer matrix

~j

q _{Jli. contains}

a combination of

temperature-dependent

vertex

weights,

the

phase diagram

of

;=i

such _row turns out to be

relatively

rich.

Let _us first consider the _q

= 2 _case characterized

by

2 _x 2

cyclic

matrices

Cl

C2 and

general Ri-,R2-matrices

written _as follows

RI ₌

Ill II)

^R2=

Iii _II) (13)

The

resulting model,

whose

weight

matrix

(4)

has the

required Z(2) (spin-flip)

symmetry

along columns,

_can coincide at certain

points

of the

weight

_space with other

models,

_e-g-,

setting

bi _{" cl " a2 "} d2 _" 0 it reduces to an 8-vertex model in _{a nonzero} field.

Inserting

^the

largest eigenvalue

of the "dominant" matrix _sum Ri + R2 into

(7),

_we arrive at

f(

_al

, a2,

bi

,

b2,_cl c2

di

d2 _"

In

(j (al

⁺ a2 +

di

+ d2 +

~/(ai

_{+ a2}

di d2)~

+

4(bi

+

b2)(ci

+

c2)) ⁽¹⁴⁾

Relation

(14)

tells _us that the system ^does not

undergo

_a

phase transition,

except in the _case with

vanishing

vertex

weights

bi _" b2 " 0 _{or cl}

" c2 " 0 when

f

₌

In(max(ai

+ _a2, di +

d2))

_~~~~

The temperature at which the first derivative of

f

with respect to the temperature is discon-

tinuous,

and _so the first-order

phase

transition takes

place,

is

given by

ai+a2"di+d2 (16)

For

general

_q, the

possibility

of

obtaining

the

explicit

results for

f((w))

and the _presence of _a

phase

transition

depend

_on the structure of

weight

matrices liq. Let _us first _suppose the

Z(q)

symmetry of vertex

weights along

_rows.

Then,

the matrices Jli. are

cyclic

and the

largest

q

eigenvalue

of their _sum

~j

_lli is

given by

;=1

q

Amax _"

~

_(1~i)3k

₍₁₇₎

I,k=1

and _so

f((w)), being

determined

by

the _sum of

positive

vertex

weights,

does not exhibit _a

singular

behaviour.

One _may expect a

phase

transition when

some of vertex

configurations along

_{rows are} not

allowed and matrices Il~ take the

block-diagonal

form. Each

diagonal

block of the matrix _sum

~

q _j

can be

diagonalized separately

and

f((w))

is determined

by

the

largest eigenvalue

Amax

;=1

q

of all blocks. Since the _non-zero elements of

£lli

include _q vertex

weights, they

_are not

simple

functions of temperature ^and Amax can ;=i'move' between blocks. The transfer of Amax

between two blocks induces the

discontinuity

in the first derivative of

f((w)).

The situation

substantially changes

when

non-vanishing off-diagonal

blocks

(on

both off-

diagonal sides)

_are taken into account. The

explicit

results for all 2 _x 2 and certain types ^of 3 _x 3 matrices ll~ indicate that

off-diagonal

blocks

destroy

the

phase

transition and

f((w))

becomes

analytical

in the whole temperature _range.

In

conclusion,

_our treatment can be

easily

extended _to

higher

dimensions. The _presence of _a symmetry of the above mentioned kind

along

_an

arbitrary

lattice direction

effectively

reduces the lattice dimension

by

_one.

References

iii

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832.

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(Academic

Press, London,

1982).

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M-T- and VAN LEEUWEN, J-M-J, Physica A 154

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365.

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