Volume 101A, number 7 PHYSICS LETTERS 9 April t984

A DILUTED QUANTUM TRANSVERSE ISING MODEL IN TWO DIMENSIONS

S. BHATTACHARYA and P. RAY Saha Institute of Nuclear Physics, 92 Acharya Prafulla Chandra Road, Calcutta- 700009, India

Received 4 January 1984

Block renormalization group method (truncation method) is applied to obtain the critical curve and exponents of the diluted zero-temperature transverse Ising model on a triangular lattice. The critical behaviour is found to be unaffected by the site dilution within the approximations of the method applied.

In recent years the application of real space renor- malization group methods, to describe the critical behav-

iour of the transverse Ising model (pure and diluted quantum spin 1/2), the so-called TIM, has been the subject of considerable theoretical interest [ 1 - 5 ] . This model has been used to analyze the nature of the phase transitions in pure and diluted order-disorder systems like cooperative Jahn-Teller systems and hy- drogen bonded ferroelectrics, which may be substitu- tionally diluted [6,7]. The zero-temperature transition (transverse field induced) o f the TIM is well studied [ 1 ] , and the fact, that the quantum critical behaviour of the zero-temperature d-dimensional transverse Ising model is the same as the thermal critical behaviour of the zero field Ising model in (d + 1) dimensions, is very often used to study the critical behaviour of higher di- mensional Ising models.

The bond diluted zero-temperature transverse Ising chain has been studied by Uzelac et al. [2], who found that the dilution does induce a cross-over to a new "random" fixed point, as is expected from Harris' cri- terion [8] applied to the diluted two-dimensional Ising model. An exact scaling of the diluted zero-tempera- ture transverse Ising chain was used by Stinchcombe [3] to obtain the exact quantum critical behaviour of the same, and the approach was also extended, by an approximate cluster method, to the diluted two di- mensional zero-temperature TIM and the result showed a discontinuity at the percolation concentration in the dependence of critical field on concentration as con- jectured by Harris [8]. However, this treatment, as

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well as the subsequent treatments [4,5] are quantita- tively not very accurate and do not indicate any insta- bility of the ground state critical behaviour o f TIM, that is expected to arise due to lattice randomness.

In this letter we treat the zero-temperature TIM on a site diluted triangular lattice, using a block renormali- zation group method (truncation method) [ 1 ], which gives more accurate results for the pure case than those of Stinchcombe [3] or of dos Santos [4]. The hamilto- nian is

' 4 4 r (1) H = - ~ J - , "'1 i

where the Pauli spin o at the ith site of a triangular lat- tice interacts only with the spin at the nearest neigh- bour site/. The entire lattice is then divided into triangu- lar blocks having three sites each (fig. 1). These blocks form a new lattice which is also a triangular one with lattice spacing V ~ times the original one.

Considering p as the probability of site occupation, the recursion relations [ 1,9] may be written as

p ' = 3 p 2 - 2 p 3 , (2)

p ' I " = p3F' y + 3p2(1 - P ) P L , (3)

p , 2 j , = p 6 j T + 6p4(1 _ p ) 2 j L , (4)

where the subscripts T and L refer to the values of the respective quantities for pure triangular lattice and pure linear chain respectively. Thus

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Volume 101A, number 7 PHYSICS LETTERS 9 April 1984

Table 1 Fixed points and critical exponents.

p* K* u z

1 2.037 0.72 1.08 0.50 0.572 0.81 1.82 3D Ising 0.63 1

Fig. 1. Spin blocks (shaded) used in the iterative procedure. "laae centres of the blocks, denoted by circles, again form a triangular lattice.

rE = ~[(4F2 + j 2 ) 1 / 2 _ j ] , (S)

JL =J (1 +a)2/2(1 + a 2 ) , (6)

where

a = (4K 2 + l ) 1/2 - 2 K , (7)

with

K = I ' /J, (8)

and

1" T = 1-' -- [(r 2 + r J + j 2 ) l / 2 _ (F2 _ I ' J + j 2 ) l / 2 ] ,

(9) JT = J (2 + a 0 + a 4)2/(3 + a 2) (3 + a 2 ) , (10)

where

a 0 = 2 ( K 2 - K + l ) - ( 1 - 2 K ) , (1 1)

a 4 = 2(K 2 + K + 1 ) - ( 1 + 2 K ) . (12)

From (2), (3) and (4)

K ' = ( 3 - 2 p ) { p [ g - ( K 2 + g + 1)1/2 + ( K 2 _ K + 1) 1/2]

+ 23-(1 - p ) [(4K 2 + 1) 1/2 - 1] }

X [ p 2 ( 2 + a 0 + a a ) 2 / ( 3 + a 2 ) ( 3 + a 2)

+ 3(1 --p)2(1 +a)2/(1 +a2)] -1 (13)

The non-trivial fixed points o f the transformations and the exponents are listed in table 1. The flow diagram has been shown in fig. 2, where the thick curve denotes the critical curve (giving the value of the critical field F/J as a function of the concentration p).

The critical exponents for pure two dimensional zero-temperature TIM obtained by employing the block renormalization method [9], are more towards the exponent values of the three dimensional thermal Ising model. Our results, therefore, are expected to be quantitatively more accurate compared to the results of earlier similar studies. The critical curve that we have obtained exhibits the expected discontinuous jump of the critical field at the percolation concen- tration. Still, the Harris instability of the pure fixed point due to lattice randomness is not yet being ex- hibited (as in ref. [3] ). This, however, is not inconsis- tent with the negative value of the exponent ~ = 2 - dv obtained in these calculations. But in this connec- tion it is worth mentioning that the treatments of dos Santos [4] and Kamieniarz [5] do not indicate the Harris instability (of pure fixed point) either, although they give positive c~ values. However, it may be noted that for diluted two dimensional Ising model also, this

2.0- )

T K

1.0

0 0.50 p -

Fig. 2. Flow lines, fixed points and critical curve (shown by the bold lines) of the site-diluted zero-temperature transverse Ising model in two dimensions.

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Volume 101A, number 7 PHYSICS LETTERS 9 April 1984

instability due to lattice randomness for cases with positive a, is not easy to get in real space renormaliza-

tion group treatments [ 10]. As already mentioned

earlier, in one dimensional diluted TIM, the extensive

study of Uzelac et al. [2] confirms this Harris instabil- ity.

We are extremely thankful to Dr. B.K. Chakrabarti

for suggesting the problem and some useful comments.

References

[ 1 ] P. Pfeuty, R. JuUien and K.A. Penson, Real-space re- normalization (Springer, Berlin, 1982)p. 119.

[2] K. Uzelac, R. Jullien and P. Pfeuty, J. Phys. A13 (1980) 3735.

[3] R.B. Stinchcombe, J. Phys. C14 (1981) L263. [4] R.R. dos Santos, J. Phys. C15 (1982) 3141. [5] G. Kamieniarz, J. Phys. C16 (1983) L1021. [6] R.T. Harley et al., J. Phys. C7 (1974) 3145. [7] A.D. Bruce and R.A. Cowley, Structural phase transi-

tions (Taylor and Francis, London, 1981) p. 93. [8] A.B. Harris, J. Phys. C7 (1974) 3082. [9] B. Hu, Phys. Lett. 71A (1979) 83.

110] M. Schwartz and S. Fishman, Physica 104A (1980) 115.

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