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On a class of extremal euclidean invariant states
Volume 29A, number 7
ON A C L A S S
P HY S I C S L E T T E R S
OF
EXTREMAL
EUCLIDEAN
16 June 1969
INVARIANT
STATES
H. J. F. KNOPS and E. J. VERBOVEN Instituut voor theoretische f y s i k a , Universiteit Nijmegen, The Netherlands
Received 13 May 1969
The notion of strongly transitive Euclidean invariant states is introduced and it is indicated that among others states with crystal symmetry occur in this class, when they are decomposed in ergodic states. The different possibilities are characterized by cluster properties.
In the a l g e b r a i c approach of quantum s t a t i s t i cal m e c h a n i c s the infinite s y s t e m is d e s c r i b e d by its C * - a l g e b r a of q u a s i - l o c a l o b s e r v a b l e s 9~. The s t a t e s of the s y s t e m a r e the positive, l i n e a r f u n c t i o n a l s on 9~ such t h a t f (e) = 1, where e i s the unit of ~ . We suppose that the Euclidean group may be r e p r e s e n t e d in the a u t o m o r p h i s m s of 9~. We a r e e s p e c i a l l y i n t e r e s t e d in s t a t e s , which a r e i n v a r i a n t for the Euclidean group, i.e. f (~g[A]) = f (A) for all A ~ 0~ and all g e E 3, where E3 stands for the t h r e e - d i m e n s i o n a l Euclidean group. We also suppose that a t i m e a u t o m o r p h i s m • t i s defined, such that the K.M.S. b o u n d a r y c o n dition is satisfied [1]. We suppose that both a u t o m o r p h i s m s ~g and ~t c o m m u t e , and that for both automorphis~ns the i n v a r i a n t s t a t e s a r e - A b e l i a n [2]. An Euclidean i n v a r i a n t state may then be decomposed in a unique way into e x t r e m a l Euclidean i n v a r i a n t states. We a r e e s p e c i a l l y i n t e r e s t e d in e x t r e m a l Euclidean i n v a r i a n t states. Such s t a t e s do not need to be e x t r e m a l i n v a r i a n t in t i m e , b e c a u s e the s y m m e t r y of the p u r e t h e r m o d y n a m i c a l phase may be lower than Euclidean s y m m e t r y . We t h e r e f o r e look for a f u r t h e r decomposition of e x t r e m a l Euclidean i n v a r i a n t s t a t e s into s t a t e s , e x t r e m a l i n v a r i a n t in t i m e (ergodic states). We i s o l a t e a s p e c i a l c l a s s of e x t r e m a l E u c l i dean i n v a r i a n t s t a t e s , which will however be l a r g e enough to i n c o r p o r a t e the p h y s i c a l l y i n t e r e s t i n g cases. Following K a s t l e r [3], we i n t r o d u c e the notion of t r a n s i t i v i t y and the r e l a t e d concepts of o r b i t and s t a b i l i z e r . The o r b i t (for the Euclidean g r o u p ) o f a state h i s the set OE of s t a t e s defined by O~ = { r t [h]; gE E3 ~, where T t is the t r a n s p o s e d a u t o m o r ~ h i s m of rg~in ~ * . Th~ s t a b i l i z e r is defined as the s u b s e t H~ o~E 3, which leaves the state h i n v a r i a n t , i.e. H~ = {geE3; r t [ h l = h}. HE is a closed subgroup of E 3. 386
An e x t r e m a l Euclidean i n v a r i a n t state f will be called t r a n s i t i v e ff the m e a s u r e ~, that dec o m p o s e s f into ergodic s t a t e s , is c o n c e n t r a t e d on one orbit,~ i.e. t h e r e e x i s t s a state h such that ~(O~) = 1. The s t a t e s o c c u r i n g the in the decomposition of a t r a n s i t i v e state r e p r e s e n t p h y s i c a l l y the s a m e pure phase, since they a r e obtained f r o m one physically pure state by an Euclidean m o v e ment. The s t a b i l i z e r H~ is ~he s y m m e t r y group of h. It is c l e a r that the s t a b i l i z e r of the state r t [hi is g-lHhE g, which m e a n s that the s t a t e s ingone o r b i t have conjugate s y m m e t r i e s . We denote by H S t h e closed subgroup of space t r a n s l a t i o n s in ~IhE. Let O ~ denote the orbit of a state h u n d e r the action of the t r a n s l a t i o n s in space above. A state f is called s t r o n g l y t r a n s i t i v e ff it is t r a n s i t i v e and ff O~ is closed for some h c support (~), where ~ is the decomposing m e a s u r e of f. R may then be proved that a t r a n s i t i v e state f is s t r o n g l y t r a n s i t i v e ff and only ff R3/H~ is compact for all h E support (lz). ( T r a n s i t i v i t y i t self is r e l a t e d with the c o m p a c t n e s s of E ° / H ~ [4]. Strongly t r a n s i t i v e Euclidean i n v a r i a n t s t a t e s may be divided into four c i a s s e s : SP 1 s t a t e s : T h e s e s t a t e s a r e a l r e a d y ergodic and the decomposition is t r i v i a l . They a r e u n i f o r m l y c l u s t e r i n g . (See ref. 4 for the definition and p r o p e r t i e s of this notion). SP 2 states: F o r these s t a t e s only rotation s y m m e t r y is b r o k e n in the decomposition of f into ergodic s t a t e s , such that H s = R 3. The r e l e vant c l u s t e r p r o p e r t y is weak mixing. SP 3 s t a t e s : Here H s i s continuous in one or two d i r e c t i o n s and d i s c r e t e in the r e m a i n i n g i n dependent d i r e c t i o n s . Rotation s y m m e t r y is then c e r t a i n l y broken. The r e l e v a n t c l u s t e r p r o p e r t y is p a r t i a l weak mixing.
Volume 29A, number 7
P HY S I C S L E T T E R S
SP4 s t a t e s : Here H s is d i s c r e t e in t h r e e i n dependent d i r e c t i o n s . Since H s i s closed and s i n c e R3/H s has to be compact, the only p o s s i b i l i t y i s that HS i s g e n e r a t e d by t h r e e n o n - c o p l a n a r t r a n s l a t i o n s , i.e. H s i s a lattice. T h i s type of states i s p h y s i c a l l y i m p o r t a n t b e c a u s e the full s y m m e t r y H~ of the ergodic s t a t e s is a space group. The r e l e v a n t c l u s t e r p r o p e r t y i s weak clustering. The SP 1 s t a t e s a r e of c o u r s e e x t r e m a l i n v a r i a n t with r e s p e c t to t r a n s l a t i o n s . All o t h e r SP s t a t e s a r e however not e x t r e m a l with r e s p e c t to space t r a n s l a t i o n s and can t h e r e f o r e be decomposed into e x t r e m a l t r a n s l a t i o n i n v a r i a n t s t a t e s [5]. It t u r n s out that the s t a t e s obtained in t h i s d e c o m p o s i t i o n obey c l u s t e r p r o p e r t i e s of different s t r e n g t h , (indicated above) which in fact e n a b l e s one to c h a r a c t e r i z e to four c l a s s e s d e s c r i b e d above. More details may be found in ref. 4. T h i s i n v e s t i g a t i o n shows that the a l g e b r a i c approach of s t a t i s t i c a l m e c h a n i c s is capable to d e s c r i b e and c h a r a c t e r i z e e q u i l i b r i u m s i t u a t i o n s with lower s y m m e t r y than the s y m m e t r y of the interaction.
16 June 1969
F r o m the m a t h e m a t i c a l point of view we i n d i cate that the i n v e s t i g a t i o n has ~been p e r f o r m e d for n o n - s e p a r a b l e C * - a l g e b r a s . I n p a r t i c u l a r it has been shown that the decomposition into e r g o dic s t a t e s on a non s e p a r a b l e C*- a l g e b r a can be p e r f o r m e d in a s a t i s f a c t o r y m a n n e r f o r t r a n s i t i v e states. [4, T h e o r e m H, 3.1 and t h e o r e m HI, 1.4].
References 1. R. Haag, N.M. Hugenholtz and M. Winnink, Comm. Math. Phys. 5 (1967) 215. 2. S. Doplicher, D.Kastler and E. St~rmer, Invariant states and asymptotic abelianess. Preprint, Marseille 1968. 3. D. Kastler, R. Haag and L. Michel, Central decompositions of ergodic states. Preprint, Marseille, 1967. 4. H. Knops, Ergodic states and symmetry breaking in phase transitions. Thesis, Nijmegen, 1969. 5. H.Knops, Comm. Math. Physics 12 (1969) 36.
* * * * *
LOCALISED
MODES
DUE
TO IMPURITIES
IN I I I - V
SEMICONDUCTORS
J. GOVINDARAJAN and T. M. HARIDASAN Department of Physics, Indian Institute of Science, Bangalore-12, India Received 21 April 1969
Local mode frequencies due to substitutional impurities in some HI-V semiconductors are calculated using Green functions on the mass defect approximation and compared with experimental results.
L o c a l i s e d m o d e s due to s u b s t i t u t i o n a l i m p u r i t i e s in m a n y of the III-V s e m i c o n d u c t o r c r y s t a l s have been o b s e r v e d in t h e i r i n f r a r e d s p e c t r a by Hayes [1], Spitzer et al.[2], Goodwin et al. [3]. The t h e o r e t i c a l c a l c u l a t i o n s of the l o c a l i s e d m o d e s in t h e s e c r y s t a l s have been made by K r i s l m a m u r t h y and H a r i d a s a n [4,5] both on the r i g i d ion approach and the shell model f o r m a l i s m b a s e d on the s i m p l e m o l e c u l a r model of J a s w a l [6]. F r o m t h e i r c a l c u l a t i o n s they have come to the c o n c l u s i o n that the m a s s defect a p p r o x i m a t i o n i s quite valid f o r the local modes in t h e s e c r y s tals. The G r e e n function approach to evaluate the local mode f r e q u e n c i e s , in the m a s s defect a p -
p r o x i m a t i o n i s n a t u r a l l y m o r e r e a l i s t i c than the m o l e c u l a r model s i n c e in the f o r m e r the e n t i r e phonon s p e c t r u m of the host l a t t i c e is made use of for the calculation. But in the case of III-V s e m i c o n d u c t o r s , no such c a l c u l a t i o n s have been attempted so far, f o r want of t h e i r complete phonon s p e c t r a . In fact, Goodwin et al. made a model calculation in the case of A1 i m p u r i t i e s in InSb, by taking the s i l i c o n d e n s i t y of s t a t e s s u i t ably modified for the host lattice. The p r e s e n t a u t h o r s have c a l c u l a t e d the phonon s p e c t r a of s e v e n III-V compounds on the b a s i s of a modified r i g i d ion model [7] and have u s e d t h e s e in the e v a l u a t i o n s of local mode v i b r a t i o n s in the G r e e n
387
ON A C L A S S
P HY S I C S L E T T E R S
OF
EXTREMAL
EUCLIDEAN
16 June 1969
INVARIANT
STATES
H. J. F. KNOPS and E. J. VERBOVEN Instituut voor theoretische f y s i k a , Universiteit Nijmegen, The Netherlands
Received 13 May 1969
The notion of strongly transitive Euclidean invariant states is introduced and it is indicated that among others states with crystal symmetry occur in this class, when they are decomposed in ergodic states. The different possibilities are characterized by cluster properties.
In the a l g e b r a i c approach of quantum s t a t i s t i cal m e c h a n i c s the infinite s y s t e m is d e s c r i b e d by its C * - a l g e b r a of q u a s i - l o c a l o b s e r v a b l e s 9~. The s t a t e s of the s y s t e m a r e the positive, l i n e a r f u n c t i o n a l s on 9~ such t h a t f (e) = 1, where e i s the unit of ~ . We suppose that the Euclidean group may be r e p r e s e n t e d in the a u t o m o r p h i s m s of 9~. We a r e e s p e c i a l l y i n t e r e s t e d in s t a t e s , which a r e i n v a r i a n t for the Euclidean group, i.e. f (~g[A]) = f (A) for all A ~ 0~ and all g e E 3, where E3 stands for the t h r e e - d i m e n s i o n a l Euclidean group. We also suppose that a t i m e a u t o m o r p h i s m • t i s defined, such that the K.M.S. b o u n d a r y c o n dition is satisfied [1]. We suppose that both a u t o m o r p h i s m s ~g and ~t c o m m u t e , and that for both automorphis~ns the i n v a r i a n t s t a t e s a r e - A b e l i a n [2]. An Euclidean i n v a r i a n t state may then be decomposed in a unique way into e x t r e m a l Euclidean i n v a r i a n t states. We a r e e s p e c i a l l y i n t e r e s t e d in e x t r e m a l Euclidean i n v a r i a n t states. Such s t a t e s do not need to be e x t r e m a l i n v a r i a n t in t i m e , b e c a u s e the s y m m e t r y of the p u r e t h e r m o d y n a m i c a l phase may be lower than Euclidean s y m m e t r y . We t h e r e f o r e look for a f u r t h e r decomposition of e x t r e m a l Euclidean i n v a r i a n t s t a t e s into s t a t e s , e x t r e m a l i n v a r i a n t in t i m e (ergodic states). We i s o l a t e a s p e c i a l c l a s s of e x t r e m a l E u c l i dean i n v a r i a n t s t a t e s , which will however be l a r g e enough to i n c o r p o r a t e the p h y s i c a l l y i n t e r e s t i n g cases. Following K a s t l e r [3], we i n t r o d u c e the notion of t r a n s i t i v i t y and the r e l a t e d concepts of o r b i t and s t a b i l i z e r . The o r b i t (for the Euclidean g r o u p ) o f a state h i s the set OE of s t a t e s defined by O~ = { r t [h]; gE E3 ~, where T t is the t r a n s p o s e d a u t o m o r ~ h i s m of rg~in ~ * . Th~ s t a b i l i z e r is defined as the s u b s e t H~ o~E 3, which leaves the state h i n v a r i a n t , i.e. H~ = {geE3; r t [ h l = h}. HE is a closed subgroup of E 3. 386
An e x t r e m a l Euclidean i n v a r i a n t state f will be called t r a n s i t i v e ff the m e a s u r e ~, that dec o m p o s e s f into ergodic s t a t e s , is c o n c e n t r a t e d on one orbit,~ i.e. t h e r e e x i s t s a state h such that ~(O~) = 1. The s t a t e s o c c u r i n g the in the decomposition of a t r a n s i t i v e state r e p r e s e n t p h y s i c a l l y the s a m e pure phase, since they a r e obtained f r o m one physically pure state by an Euclidean m o v e ment. The s t a b i l i z e r H~ is ~he s y m m e t r y group of h. It is c l e a r that the s t a b i l i z e r of the state r t [hi is g-lHhE g, which m e a n s that the s t a t e s ingone o r b i t have conjugate s y m m e t r i e s . We denote by H S t h e closed subgroup of space t r a n s l a t i o n s in ~IhE. Let O ~ denote the orbit of a state h u n d e r the action of the t r a n s l a t i o n s in space above. A state f is called s t r o n g l y t r a n s i t i v e ff it is t r a n s i t i v e and ff O~ is closed for some h c support (~), where ~ is the decomposing m e a s u r e of f. R may then be proved that a t r a n s i t i v e state f is s t r o n g l y t r a n s i t i v e ff and only ff R3/H~ is compact for all h E support (lz). ( T r a n s i t i v i t y i t self is r e l a t e d with the c o m p a c t n e s s of E ° / H ~ [4]. Strongly t r a n s i t i v e Euclidean i n v a r i a n t s t a t e s may be divided into four c i a s s e s : SP 1 s t a t e s : T h e s e s t a t e s a r e a l r e a d y ergodic and the decomposition is t r i v i a l . They a r e u n i f o r m l y c l u s t e r i n g . (See ref. 4 for the definition and p r o p e r t i e s of this notion). SP 2 states: F o r these s t a t e s only rotation s y m m e t r y is b r o k e n in the decomposition of f into ergodic s t a t e s , such that H s = R 3. The r e l e vant c l u s t e r p r o p e r t y is weak mixing. SP 3 s t a t e s : Here H s i s continuous in one or two d i r e c t i o n s and d i s c r e t e in the r e m a i n i n g i n dependent d i r e c t i o n s . Rotation s y m m e t r y is then c e r t a i n l y broken. The r e l e v a n t c l u s t e r p r o p e r t y is p a r t i a l weak mixing.
Volume 29A, number 7
P HY S I C S L E T T E R S
SP4 s t a t e s : Here H s is d i s c r e t e in t h r e e i n dependent d i r e c t i o n s . Since H s i s closed and s i n c e R3/H s has to be compact, the only p o s s i b i l i t y i s that HS i s g e n e r a t e d by t h r e e n o n - c o p l a n a r t r a n s l a t i o n s , i.e. H s i s a lattice. T h i s type of states i s p h y s i c a l l y i m p o r t a n t b e c a u s e the full s y m m e t r y H~ of the ergodic s t a t e s is a space group. The r e l e v a n t c l u s t e r p r o p e r t y i s weak clustering. The SP 1 s t a t e s a r e of c o u r s e e x t r e m a l i n v a r i a n t with r e s p e c t to t r a n s l a t i o n s . All o t h e r SP s t a t e s a r e however not e x t r e m a l with r e s p e c t to space t r a n s l a t i o n s and can t h e r e f o r e be decomposed into e x t r e m a l t r a n s l a t i o n i n v a r i a n t s t a t e s [5]. It t u r n s out that the s t a t e s obtained in t h i s d e c o m p o s i t i o n obey c l u s t e r p r o p e r t i e s of different s t r e n g t h , (indicated above) which in fact e n a b l e s one to c h a r a c t e r i z e to four c l a s s e s d e s c r i b e d above. More details may be found in ref. 4. T h i s i n v e s t i g a t i o n shows that the a l g e b r a i c approach of s t a t i s t i c a l m e c h a n i c s is capable to d e s c r i b e and c h a r a c t e r i z e e q u i l i b r i u m s i t u a t i o n s with lower s y m m e t r y than the s y m m e t r y of the interaction.
16 June 1969
F r o m the m a t h e m a t i c a l point of view we i n d i cate that the i n v e s t i g a t i o n has ~been p e r f o r m e d for n o n - s e p a r a b l e C * - a l g e b r a s . I n p a r t i c u l a r it has been shown that the decomposition into e r g o dic s t a t e s on a non s e p a r a b l e C*- a l g e b r a can be p e r f o r m e d in a s a t i s f a c t o r y m a n n e r f o r t r a n s i t i v e states. [4, T h e o r e m H, 3.1 and t h e o r e m HI, 1.4].
References 1. R. Haag, N.M. Hugenholtz and M. Winnink, Comm. Math. Phys. 5 (1967) 215. 2. S. Doplicher, D.Kastler and E. St~rmer, Invariant states and asymptotic abelianess. Preprint, Marseille 1968. 3. D. Kastler, R. Haag and L. Michel, Central decompositions of ergodic states. Preprint, Marseille, 1967. 4. H. Knops, Ergodic states and symmetry breaking in phase transitions. Thesis, Nijmegen, 1969. 5. H.Knops, Comm. Math. Physics 12 (1969) 36.
* * * * *
LOCALISED
MODES
DUE
TO IMPURITIES
IN I I I - V
SEMICONDUCTORS
J. GOVINDARAJAN and T. M. HARIDASAN Department of Physics, Indian Institute of Science, Bangalore-12, India Received 21 April 1969
Local mode frequencies due to substitutional impurities in some HI-V semiconductors are calculated using Green functions on the mass defect approximation and compared with experimental results.
L o c a l i s e d m o d e s due to s u b s t i t u t i o n a l i m p u r i t i e s in m a n y of the III-V s e m i c o n d u c t o r c r y s t a l s have been o b s e r v e d in t h e i r i n f r a r e d s p e c t r a by Hayes [1], Spitzer et al.[2], Goodwin et al. [3]. The t h e o r e t i c a l c a l c u l a t i o n s of the l o c a l i s e d m o d e s in t h e s e c r y s t a l s have been made by K r i s l m a m u r t h y and H a r i d a s a n [4,5] both on the r i g i d ion approach and the shell model f o r m a l i s m b a s e d on the s i m p l e m o l e c u l a r model of J a s w a l [6]. F r o m t h e i r c a l c u l a t i o n s they have come to the c o n c l u s i o n that the m a s s defect a p p r o x i m a t i o n i s quite valid f o r the local modes in t h e s e c r y s tals. The G r e e n function approach to evaluate the local mode f r e q u e n c i e s , in the m a s s defect a p -
p r o x i m a t i o n i s n a t u r a l l y m o r e r e a l i s t i c than the m o l e c u l a r model s i n c e in the f o r m e r the e n t i r e phonon s p e c t r u m of the host l a t t i c e is made use of for the calculation. But in the case of III-V s e m i c o n d u c t o r s , no such c a l c u l a t i o n s have been attempted so far, f o r want of t h e i r complete phonon s p e c t r a . In fact, Goodwin et al. made a model calculation in the case of A1 i m p u r i t i e s in InSb, by taking the s i l i c o n d e n s i t y of s t a t e s s u i t ably modified for the host lattice. The p r e s e n t a u t h o r s have c a l c u l a t e d the phonon s p e c t r a of s e v e n III-V compounds on the b a s i s of a modified r i g i d ion model [7] and have u s e d t h e s e in the e v a l u a t i o n s of local mode v i b r a t i o n s in the G r e e n
387