Second - o r d e r phase transitions are usually descr i b e d in the framework of the r e n o r m a l i z a t i o n group c a l c u l a t i o n s in terms of fixed points which are accessible on the critical surface. The situation is not so clear for firs t - o r d e r
transi-2 4 transi-2 5
tions. There is a large class of magnetic systems ' , where the first-order transition has been attributed to the absence of stable fixed points w ithin the domain of s t a b i l i t y of the model. This does not apply here, since there is always an accessible fixed point for any value of q.
2 6
N i e n h u i s and N a u e nberg have introduced an important new concept, the d i scontinuity fixed point. There are systems
where the first-order transition can be described in terms of the b e h a v i o r around a special fixed point. A suf f i c i e n t c o n d i tion to get a discont i n u i t y in the order p a r a m e t e r or other d e r i v a t i v e s of the free e nergy is to have a fixed point, where the thermal and magnetic eigenvalues of the RG t r a n s f ormation are equal to b ^ , where b is the scale factor in the RG transfor mation and d is the dimensionality. A fixed p o i n t where the e i g envalues are equal to b^ is called d i s c o n t i n u i t y fixed p o i n t
The calculated values of the eigenvalues are given in Table II. As one can see both the thermal and magn e t i c eigen-values tend to 2 as q “ but they are not equal to b d for any finite value of q. Nevert h e l e s s one can argue, that the fixed points, which become accessible for q>q^ » are d i scontinuity
fixed p o int and describe a first-order transition.
f
VI. DISCUSSION
In this paper we have p r e s e n t e d a RG treatment of the
qua n t u m version of.the 2-d Potts model. Our RG c a l c u l a t i o n gives a crossover from second-order to first-order tr a n s i t i o n as the number of components increases, in agreement with the known exact results. The fixed p o int structure of our RG transfor-mation is very similar to that obtained by Nienhuis et a l .13 The r e n ormalized H a m iltonian has two couplings, h/A and x, the latter one being generated d u r i n g the r e n o r m a l i z a t i o n only.
Below a certain critical value of the n umber of components, q , c there are three fixed points, two of them at finite values of the couplings, one at infinity. The two fixed p oints at finite values merge at q^ and annihilate each other and th e r e b y the infinite fixed p o int becomes p h y s i c a l l y accessible. At this fixed p o i n t the specific heat exponent a = 1 and we argued that this is an indication of the first-order nature of the transition.
We want to emphasize that our RG tran s f o r m a t i o n is a standard one and no ambiguity was built in, contrary to the m o d i f i e d major i t y rule of the vacancy model of Nienhuis et al.'*'^.
Our r e n o r m a l i z a t i o n group t r a n s f ormation g e n e r a t e d a new coupling x / 1. The coupling was introduced in the r e p r e s e n tation defined by the states of E q . (2.9). In terms of the
original Potts states the new coupling will flip s i m u l t a n e o u s l y two neighb o u r i n g spins. One can try to construct a classical 2-d model whose transfer m a t r i x would c o r respond to our r e n o r m a lized Hamiltonian. This classical model will contain four-spin
coupling on a plaquette, but the strengths of the possible four-spin terms are all re l a t e d to each other. It seems that the i n t roduction of the four-spin c o u p l i n g s has the same effect as the i n t r o d u c t i o n of v a c a n c y variables and allows to describe the crossover fr o m second-order to f i r st-order transition.
The value o b t a i n e d for 4 C ' 9 C = 6.81, is somewhat far from the exact r e s u l t = 4. This is the consequence of taking two sites per cell and k e e p i n g for eac h cell the q lowest lying states only. Improvements c o u l d be o b t a i n e d either by taking larger cells or by taking into account the higher lying states
1 6 in a p e r t u r b a t i o n a l way as p r o p o s e d for the Ising model by Urn
, . 17
and H i r s c h and M a z e n k o . In the first case d i a g o n a l i z a t i o n of large matrices w o u l d be needed, while in the second case an extremely large n umber of n e w couplings would be generated.
We feel that both methods w o u l d give a slight impr o v e m e n t of q c , leaving our conclusion on the c r o s s o v e r from second-order transition to first-order intact.
ACK N O W L E D G E M E N T S
One of us (J.S.) is grateful to E. Fradkin for many useful d i s cussions and to R. Pandit and M. Ma for their help in the numerical calculations.
REFERENCES
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D. Kim and R.I. Joseph, J. Phys. A8^, 891 (1975) . 6. M. Yamashita, Progr. Theoret. Phys. 6_1, 1287 (1979) . 7. V.M. Zaprudskii, Zh. Eksp. Teor. Fiz. 73 , 1174 (1977) ;
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R.K.P. Zia and D.J. Wallace, J. Phys. A8, 1495 (1975);
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A£, L I 79 (1976) ; B.W. Southern, J. Phys. A10, L253 (1977) . 13. B. Nienhuis, A .N. Berker, E.K. Riedel and M. Schick,
Phys. Rev. Lett. 4_3 , 737 (1979) .
14. S. Jafarey, R. Pearson, D.J. Scalapino and B. Stoeckly
( u n p u b l i s h e d ) : S.D. Drell, M. W e i n s t e i n and' S. Yankielowicz Phys. Rev. Dl£, 1769 (1977) .
15. R. Juliién, P. Pfeuty, J.N. Fields and S. Doniach, Phys. Rev. В П Ь 3568 (1978) .
16. G. Urn, Phys. Rev. B 1 5 , 2736 (1977) .
17. J.E. Hirsch and G.F. Mazenko, Phys. Rev. Bl_9, 2656 (1979) . 18. A. F e r n a n d e z - P a c h e c o , Phys. Rev. В^Э, 3173 (1979) .
19. E. Fradkin and S. Raby, Phys. Rev. D2C), 2566 (1979).
20. L. Mittag and M.J. Stephen, J. Math. Phys. 1^, 441 (1971).
21. M.J. Stephen and L. Mittag, Phys.Lett. 4 1 A , 357 (1972). t 22. J. Kogut, Rev. Mod. Phys. 51_, 659 (1979) .
23. J. Sólyom, Phys. Rev. following paper. •
24. S .A . Brazovskii and I.E. D z y a l o s h i n s k i i , Pisma Zh. Eksp.
Teor. Fiz. 2_1, 360 (1975) ;
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FIGURE CAPTION
Fig. 1. Locations of the fixed p o i n t s on the h/Ax 2 versus 1/x plot. q is a p a r a m e t e r of the curves. Fixed points on the straight lines can be reached from the x = 1 Potts line. The fixed points on the dotted lines are accessible for a more general model only.
CAPTION OF TABLES
Table I. The thermal and magnetic e i g e n v a l u e s of the RG transformation and the critical e x p o n e n t s of the q-state Potts model, obta i n e d by a self-dual RG t r a n s f o r m a t i o n .
Table II. The critical and fixed-point values of the couplings obtained with a block tran s f o r m a t i o n for several q values and the thermal and magnetic eige n v a l u e s of the RG transformation.
Table III. The critical e x p o nents obta i n e d with a block t r a n s formation with two sites per block.
eigenvalue s
q
c r i t i c a l c o u p l i n g
h A
f i x e d p о i n t e i o e n v a l u e я
h
“ ’ 4
X
P"T h
t h e r m a l X t
ma gne tic Am
x2 1 2
2 0 . 6 3 8 0 . 6 3 8 1 . 0 0 0 0 . 6 3 8 1.5 9 6 1.652
3 0 . 3 7 3 0 . 3 9 2 1. 1 1 5 0 . 3 1 5 1.7 7 8 1 . 6 3 5
4 0 . 2 6 0 0 . 2 9 7 1 . 1 9 3 0 . 2 0 9 1.966 1 . 6 2 1
5 0 . 1 9 9 0 . 2 5 8 1 . 3 0 7 0 . 1 5 1 2 . 1 7 7 1.6 0 7
6 0 . 1 6 1 0 . 2 5 7 1 . 5 1 3 0 . 1 1 3 2 . 4 5 5 1.5 9 2
6 . 8 1 0 . 1 3 9 0 . 4 0 7 2 . 2 5 3 0 . 0 8 0 3 . 0 2 0 1 . 5 6 7
6.82 0 . 1 3 9 00 oo 0 . 0 6 0 3.184 1 . 5 8 5
7 0 . 1 3 5 0 0 o o 0 . 0 6 0 3.1 2 5 1 . 6 0 0
8 0 . 1 1 6 0 0 0 0 0 . 0 6 0 2 . 8 8 0 1.666
9 0 . 1 0 2 00 0 0 0 . 0 5 9 2 . 7 2 2 1.7 1 4
10 0 . 0 9 1 0 0 0 0 0 . 0 5 7 2 . 6 1 2 1 . 7 5 0
СО 0 0 o o
0 2 2
q V -1
у = V z z V
2 1 . 4 8 3 0 . 6 7 4 0 . 5 5 1 0 . 8 1 8
3 1 . 2 0 5 0 . 8 3 0 0 . 5 8 1 0 . 7 0 0
4 1.025 0 . 9 7 5 0 . 6 0 7 0 . 6 2 2
5 0 . 8 9 1 1.122 0 . 6 3 1 0 . 5 6 3
6 0 . 7 7 2 1.295 0 . 6 5 9 0 . 5 0 8
6 .81 0 . 6 2 7 1.594 0 . 7 0 5 0 . 4 4 2
6 .82 0 . 5 9 8 1 . 6 7 1 0 . 6 7 1 0 . 4 0 1
7 0 . 6 0 8 1.6 4 5 0 . 6 4 4 0 . 3 9 2
8 0 . 6 5 5 1 . 5 2 7 0 . 5 2 6 0 . 3 4 4
9 0 . 6 9 2 1.4 4 5 0 . 4 4 5 0 . 3 0 8
10 0 . 7 2 2 1.3 8 5 0 . 3 8 5 0 . 2 7 8
oo 1 1 0 0
V A ß/v ß a
0 . 8 2 0 . 2 7 6 0 . 4 0 9 - 0 . 3 0 1
0 . 7 0 0 . 2 9 1 0 . 3 5 0 0 . 0 9 5
0 . 6 2 0. 303 0 . 3 1 1 0 . 3 5 3
0 . 5 6 0 . 3 1 6 0 . 2 8 1 0 . 5 4 6
0 . 5 1 0 . 3 2 9 0 . 2 5 4 0 . 7 2 0
0 . 4 7 0 . 3 5 2 0 . 2 2 1 0 . 9 3 1
0. 336 0 . 2 0 1 1
0 . 3 2 2 0 . 1 9 6 1
0 . 2 6 4 0 . 1 7 2 1
0 . 2 2 3 0 . 1 5 4 1
0.2 6 0 . 1 9 3 0 . 1 3 9 1
0 0 1
Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Krén Emil
Szakmai lektor: Zawadowski Alfréd Nyelvi lektor: Zawadowski Alfréd Példányszám: 520 Törzsszám: 81-14 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly
Budapest, 1981. január hó