The Potts spin-glass in a transverse field

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The Potts spin-glass in a transverse field

Yadin Goldschmidt

To cite this version:

Yadin Goldschmidt. The Potts spin-glass in a transverse field. Journal de Physique I, EDP Sciences,

1992, 2 (1), pp.31-39. �10.1051/jp1:1992116�. �jpa-00246460�

Classification Phydcs Abstracts

05.20 05.30 75.50L

The Potts spin-glass in

_a

transverse field

Yadin Y. Goldschmidt

Department of Physics and Astronomy, University ofPittsburgh,

Pittsburgh,

PA 15260, ^U-S-A-

(Received

23 July 1991, accepted 20

September 1991)

Abstract. ln this _{paper we} derive _an expression for the free _energy of the infinite ranged quantum ^Potts spin-glass in

a transverse field using the replica method and the Suzuky-Trotter

formula. Results _are obtained for general _p

(number

of Potts

states)

and explicit calculations

are done for _p

= 3

using

exact spin summations for M _< 8 where M is the size of the lattice in the Trotter direction. The phase diagram is obtained and is also compared with results for the

static approximation which is _non trivial _even in the paramagnetic phase.

I Introduction.

The

investigation

of the quantum

spin-glass

model attracted much attention

recently [1-17].

It has been found that quantum

tunneling

has _a

disordering

ef§ect similar to the temperature.

In the _case of the

Ising

model in _a transverse

field,

the transverse field simulates the effect of

tunneling

in proton

glasses

[3]. ^{It has been found that this}

field,

when strong

enough, destroyes

the

spin- glass phase

_even at T

= 0~ but the nature of the

spin-glass phase,

when it

exists,

for the _case of the infinite _range

model,

is _very similar to the classical _case

(r

₌

0)

and is characterized

by

a Parisi order paramater

q(z) (or alternatively by

_{a non} trivial distribution of

overlaps P(q)) [18]. Experimental

realizations of proton

glass

_occur in random mixtures of ferroelectric and antiferroelectric materials [3], ^{and it has been}

proposed recently

that the

quantum Heisenberg spin-glass

_may

play

_a role for _{some range} of

doping

in the

newly

discovered

high-n

materials

(in

the

insulating phase)

_[19].

The classical infinite

ranged

Potts

spin glass

^has some

interesting

features which have been

investigated

in recent years [20,

21].

In

particular

for _{p >} 4

(p

is the number of Potts

states)

the

spin-glass

transition has _some first order

features,

and for _{any p} > 3 the order parameter

q(z)

has _a

particularly simple

form for _{some range} of temperatures ^below Tg [21].

In this _{paper we}

generalize

the

Ising

model in _a transverse field _to the _case of the Potts model with

general

_p and _use the

replica

method and the

Suzuky-Trotter

formula

[22,

23] to map the model into _an

equivalent

classical system. ^The Potts model in _{a transverse} field which represent

tunneling

^{between the} _p

equivalent

Potts states is

quite

_non ^trivial. Even the static

approximation

in the

paramagnetic phase

is difficult to

implement

for

general

_p.

32 JOURNAL DE PHYSIQUE I N°1

After

deriving

_some

general

results _we concenrate _on the _{case p}

= 3 where the transition is

expected

to be of second order for _any value of the transverse field r

(we verify

this

explicitly

for M =

2).

We first determine the location of the

phase boundary using

the static

approximation.

We then _carry out exact

spin

summations _{up to} M < 8 and

extrapolate

the results _to M _{= oc}

using

the

I/M~

law [24]. ^{The result obtained}

by

this method for the

rc(T

₌

0)

^{is found} to be lower than static

approximation

result which is known from the

Ising

_case to over-estimate the value of the transverse field _necessary to

destroy

the transition at T

= 0.

Finally,

We

point

out

questions

and directions for future research.

2. The free energy and order paranJeters for

general

_p.

Following

the discussion in reference [25] we consider the

following

Hamiltonian for the Potts model in _a transverse field:

p-I p-I

~f ₌

~

_j,,

£(u±u,)m r£ ~

vp

(i)

iJ _i J i

(ij) ^m=0 I m=0

Here

(ij)

is _a summation _over all distinct

pairs

of

sites,

and it is _a summation _over all sites.

V and U are site variables

(operators)

which _{on a}

given

site

satisfy

the

Zp algebra

VU =

e~~~'/PUV, V+U

=

e~~'/PUV+,

VP

= UP

=

(2)

In _a

particular representation

in which U is

diagonal

_one has

U

nj

₌

e~'+ nj,

n = 0,

, p

(3j

V

n) =I

n +

ii

and

PI

₌ 0)

(4)

Jij

_are random variables which for

simplicity

_can be chosen to

satisfy

_a ^Gaussian

probability

distribution of width

J/@

and _mean Jo

IN

where N is the total number of sites.

We _now make _use of the Suzuki-Trotter formula to map the system into _a classical system

plus

_an extra '~Trotter" dimension. The effective classical Hamiltonian for the Mth

approxi-

mant is

given by

~

M N

~~~~~~ ~

M

( ( ~"~~'l~~>~J(~)

^{~ ~}

~ $ ~n,(k),n,(k+i)

+

CNM, (5)

with

~ ~

B ₌

In(

^~_~'~j~^~

),

^C ₌

In(

^{~ ~'}

(6)

e M I P

and ₇

=

fly.

In

deriving equation (5)

we have used the identities

exp(7 f

V~)

= 1+ ^~~~

f V~' (7)

m=0 P

m=0

(n exp(7 f

V~) n')

=

bn,n,

+

~~~

(8)

m=0 P

where the

eigenstates n)

of U where defined in

equation (3).

In

equation (5)

^the classical variables

n,(k)

take the values

0,

_p- I _on sites

(I, k)

where k =

I,

,

M labels the Trotter direction and

satisfy

the

periodic boundary

condition in this

"time"

direction. _Ultimately

_{the limit M}

- oc must be taken. The

quenched averaging

_over

the random bonds is done

by

the _use of the

replica

trick

[26].

^After some

algebra

_we obtain:

"

_"

4flln 11 £iQ17'(k> k')12

+

t _{~o ~o} ~imz<~,j~

no ^, ~~ ~ ~ ~

-(In~hexp(G) ⁽⁹⁾

with

p2 j2

~ _"

~ _£(~~na

,

(k),na(k+i) ⁺

tl)

+

~ 2 Jf2

~ ~ ~ _Qll

_(~>

k')(P~n"(k),~ l)(P~n«'(k'),s

^~)

a i P

««, ;1, _»

+

jjj ⁱⁱ LMI(k)(p&n«~i~,~ ^{i) (lo)}

~ ~ i

where the linfit

n - 0 has _to be taken. We _now define

j~a

(~ ~l) Qaa(~ ~l) (~~)

~8 ~8 '

and _reserve the notation

Qff'

_{for the}

case where _a

#

o'. Based _on the Potts symmetry ^of the model _we look for

stationary

solutions

satisfying

Q(f'

=

Q°°'(pbm

₁₎

(12)

Rls

₌

R°(pb~s

₁₁

(13)

We will further

simplify

the

expression

for the free _energy

by making

the

following

_assump-

tions and arguments:

First since _{we are} interested

only

in the

spin-glass phase

and not in _a

ferromagnetic phase (or

a mixed

phase)

_we choose

Jo

to be

negative

such that

Mf(k)

₌ 0. This choice has also

been used in the classical _case

[20, 21].

Next _{we assume} that

Q°°'(k, _k')

does _not

depend

_on the Trotter index. This has also been

used in the

Ising

_case and follows from the _nature of

Q

_as the

spin-glass

order parameter, and it is found to be _a consistant solution of the

stationarity

conditions. For

R°(k, k')

we will _assume

that it does not

depend

on a since there is _no

prefered

direction in

replica

_space and that the

dependence

_on

k,

k' _comes

through

the difference k k'

(because

of translation invariance in the Trotter

direction).

To

simplify

matters _we will _{use a} first step

replica

symmetry

breaking

scheme [27] ^{in the}

spin glass phase

which for the classical _case

(for

_p >

2)

is known to be exact in _a range of temperature ^below

Tg.

[21]

Thus _one puts _{a =}

(L7)

with L ₌ I,

,

n/m;

_{7 "} 1, m and

Q°°'

"

Q2

L

"

L'>'f # 'f' ₍₁₄₁

Q°°'

=

Qii

L

#

L'

(15)

In the classical _case for _{p >} 2 _one has

Qii

_" 0 in _{a range} of temperatures ^below

Tg.

^Since it is _not

guaranteed

to hold for r

#

0 _we will

keep

the _most

general

_case

(15).

First step ^RSB

34 JOURNAL DE PHYSIQUE I N°1

even if not exact is _a

good

first

approximation

to the

spin glass

_{case as} has been found for the

Ising spin glass

in _a transverse field [14]

(p

₌

2).

We will _{now use} the Wallace-Zia [28]

representation

of the Potts variables _{as p} I dimen- sional vectors

S(n),

_{n =}

0,

,p I.

We have

pbnn,

I ₌

S(n) S(n') (16)

The free _energy

can now be written in the form

p2

j2

~ i

~~

~

~ ~~

~~~~~~2 ~~l) ~(

₊

j~2

~~~~

~ ~' ~

ii'

jDP~~z

ⁱⁿ

jDP~~y ^{Z'$ (17)}

m

~~~~~

~p-i, jj

_~~'~

)exp(- £ z?/2) _(~~)

@

^~

n>=i I

and

similarly

for

DP~~y. ZH

is

given by

Tt

exp(H)

where

H ₌

j _L(R(I

k k'

1)

Q2)S(n(k)) S(n(k'))+

+

~(

^~

_{S(n(k)) S(n(k}

₊

₁₎₎

₊ ^~ ₊

_C)

~ P P

+$( _Qii

y +

/& _z) ~ _{s(n(k)) (19)}

from

equations (17)-(19)

_we obtain the

stationarity equations

for the R's and

Q's

q

_ ^-l~

ji~

(~ij

~~

p

- I

I _~

^~yZg

where

(A)H

_"

lh[A exp(H)]/lh exp(H) (23)

In the

pararnagnetic phase equation (17)

becomes

Pi

₌

(jj~ ^(p

₁₁

~ _R(

k k'

1)~ in zH

(24)

~ ~

kk,

with

H =

~ _£

R(I

^{k k'}

I>S(n(k» S(n(k'l>

₊

L()S(n(k» ^S(n(k

⁺

^ii>

⁺

)

+

Cl (251

and

R~i

^{k k'} _{i> "}

)iS<n<k» S<n<k'»iH

_(26>

In the _case that the transition is continuous

<which

^for _q < 4 is known to be true when r

=

0),

then the

phase boundary

^{is determiend}

by

the condition This is found

by expanding

^the

free _energy in _powers of

Q

and

identifying

the coefficient of the

quadratic term>:

ji _£

~<

^{k k'} _I> "

TC/J

_<27>

k,k'

For the _case when the transition is of first order

equation

_<27> is _no

longer

valid since the location of the transition is _no

longer

determined

by vanishing

^{of the} coefficient of the

Q~

term in the

expansion

^{of the free} _energy in _power of

Q.

In that

case the location of the transition

can be found

by matching

^the free

energies

in the

paramagnetic

^{and in} the

spin-glass phases.

3.

Explicit

calculations for _p

= 3.

In this section _we will _carry out

explicit

calculations for the three state Potts model. We will

assume that _as in the classical _case with r ₌

0,

the transition is continuous and hence _{we can}

use the condition

(27>.

We have verified that this is the _case for M ₌ 2

(see _below>

where _we

were able to calculate the order parameters

Q2

and

Qii

in the

spin-glass phase.

Before _we

proceed

with the _exact calculations _we will

briefly

^{describe the results} of the static

approximation.

In the static

approximation R(k,

_k'> is taken to be

independent

of

k,

k'. We then find that

equation

_(24> taken the form

fl f

₌

~~~~

R~

₇ lim in G <28)

with

G

=

j

_{Dz 'h}

_{exPl~ £(S(n(kl S(n(k+ Ill 2>+} $4

z

.~lS<n<k»

+

S<n(k+ I»11

_(29>

s(o>

₌

<v5,o>, s<i>

=

<-(, _{(>, s<2)}

=

<- (, _(>

<30)

The trace in

equation (29)

_can be obtained from the

eigenvalues

of _a 3 _x 3 transfer matrix

representing

_a I dimensional Potts model. Because of the M

dependence

of B B

--

In(M/r)

for

large M)

and of the

magnetic

field _one needs _to consider _{a non-} trivial M

- oc limit.

This _means that _one has _to calculate the

I/M

correction _to the

eigenvalues li

of the transfer

matrix.

Pitting

li ₌ I +

flui /M

+

O(I /M~

_we find

G =

j

_Dz

_£efl~'

<31)

I

l

with _u,

being

solutions of the

equation

u~ 3<(h( ^I )h(

⁺

^{r~)u 2r~} #h(

₊

_~#hi hi

= 0 <32>

36 JOURNAL DE PHYSIQUE I N°1

h;

=

J4

z;

(33)

Equation (33)

_can be solved

numerically

and

together

with

equations (28) (31)

and the condition R ₌

I/flcJ

_can be used to find the

phase boundary

between the

spin-glass

and

paramagnetic phases.

^Here we consider the

limiting

_cases of r

= 0 and T ₌ 0.

In the _case r ₌ 0 _one

easily

find

~~~

"~

~~~'

"~

~~~

~

~~~'

~~

~~~ ~~~

~~~~

p j

₌

1p2 j2

j~2

p2 j2

_{j~ f~ ~}

(~~j

~ from which R

= I and also Tc ₌ J. This _agrees with known results for the classical _case.

In the _case Tc _- 0 _we would like to find r

(Tc)

in that limit. We have to consider the limit flc _- oc and R

- 0

(since

Rc

=

I/flcJ

at the

transition)

and thus h; in

equation (32)

_are also small. Under these conditions the

largest

solution for

equation (32)

is

um = 2r +

(h(

₊

h()

₊

= 2r +

J~Rz~

₊

(36)

3r 3r

Thus the free _energy is

given

in the small T limit

by Pi

₌

(~~

R2

3Pr

₊

in(1 ~((~ ₍₃₇₎

From this

equation

and the condition ll~ ₌ I

/flcJ

_we find

rc(T

"

0)

_"

~J ⁽³⁸⁾

In _case of the

Ising

model

(p

₌

2)

it is known that the static

approximation

tends _to overestimate the value of rc at T =0.

(The

exact value is about

rc(0)

-- 1.51J whereas the static

approximation yields

rc (0> =

2J>.

We _now

proceed

to _an exact evaluation of the location of the

phase boundary using

exact

spin

summations. We have evaluated the free _energy

(24)

and solved for R (( ^k k' for lattices with the size of the Trotter dimensions _up to M ₌ 8

(with periodic b-c-).

For M ₌ 2

we have also solved for the solution in the

spin- glass phase.

In the

spin-glass

_one has to carry out the two dimensional

integrals

d~z and

d~y

in

equations <20)-(22)

and the solution is

quite

time

consuming.

The results of the

spin-glass phase

calculation

indicates,

at least for M

= 2,

that:

(a)

^The transition is of second order for _p

= 3 _even for r

#

0;

(b)

The first stage

replica breaking

with

Qii

" 0 is _a solution of the

stationarity

condition

for _a range of temperatures ^below

Tc<r)

_as it is for r

= 0.

More work has to be done to establish these results for M _{= oc}

although

the results _seems

plausible. Assuming

the transition to be continuous _we derived the location of the

phase boundary

from the solution of

equations (26)

and

(27).

For temperatures 0.95, 0.9, 0.8, 0.6 _we have calculated

rc(T>

for M

= 2, 4, 8. For T

= 0A _we evaluated

rc<T>

^{for M} =

4,8.

The values obtained for

rc(T)

_{as a} function of

I/M~

lie _on

nearly straight

^lines.

(This

is the known

I/M~

law

[24])

^and

by

this method _we

extrapolated

to M

= cxJ

(allowing

for _some small _error

bars)).

The results _are

displayed

in

figures

I and

figure

2.

Figure

2 shows the location of the

phase boundary

and _{we see} that

rc(T

=

0)

-- 1.05- 1-1This is

compared

with the value of

4/3

obtained from the static

approximation

which is known _to overestimate the value of

rc(T)

for small T.

2.0

Legend 1.8

. T=0.g5

x T=O.g

6 _a T=O.8

. T=0.6

1_4 ~"~'~

l.2

~ i.o

0.8

O.6

O.4

O.2

0 O.025 O.OSO 0.075 0.100 O.125 0,150 O,175 0.200 0.225 O.250

1/M62

Fig. 1. rc _~3.

1/M~

foT various values of T for the _p

= 3 Potts SG.

1.2

1-o

0.8

~

O.6

O.4

, '

,

O.2 ,

0

0 0.2 0.4 O.6 0.8 1.0 .2

r

Fig. 2. Phase boundary between the

Spin-glass

and paramagnetic phases for the

case p = 3. The

dotted line is _a possible extrapolation to T

= 0.

38 JOURNAL DE PHYSIQUE I N°1

4. Conclusions.

In this _{paper we} extended the

Ising spin-glass

in _a transverse field to _a p-component Potts model with _{a transverse} field that induces

tunneling

_among the p-states. This model _may have

experimental

realizations in ferroelectric antiferroelectric mixtures and is also

interesting

from the theoretical

point

view. More work is needed to

investigate higher

_p

values, especially

_p >

4, where the transition is known _to have _some first order features for r ₌ 0

[21].

A solution for _p

- oc may also be found. These _are

projects

for future research.

Acknowledgements.

This work has been

supported by

the National Science Foundation under grant

#

DMR- 90l6907.

Note added in proofs: After the

completion

of this w6rk _we became _aware of _a

preprint by

K. Walasek and K. Lukierska-Walasek who consider the 3 state Potts

glass

in _a transverse field.

They

do not _use

replicas,

but rather _an uncontrolled effective field

approximation

that

neglects

correlations _among different components of the

cavity

field. Their result of

rc(0)

₌ 2

/3

underestimates the value of the critical field at T

= 0.

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