Volume 131B, number 1,2,3 PHYSICS LETTERS 10 November 1983

LATTICE DEGENERACIES OF GEOMETRIC FERMIONS

H. R A S Z I L L I E R Physikalisches lnstitut, Universitiit Bonn, Nussallee 12, D-5300 Bonn 1, West Germany

Received 2 June 1983 Revised manuscript received 1 July 1983

We give the minimal numbers of degrees of freedom carried by geometric fermions on all lattices of maximal symmetries in d = 2, 3, and 4 dimensions. These numbers are lattice dependent, but in the (free) continuum limit, part of the degrees of freedom have to escape to infinity in the spectrum by a Wilson mechanism built in, and 2 e survive for any lattice. On self-reciprocal lattices we compare the minimal numbers of degrees of freedom of geometric fermions with the minimal numbers of naive fermions on these lattices and argue that these numbers are equal.

R e c e n t l y the re has b e e n an increas ing in ter - est in the fo rmu la t i on and inves t iga t ion of lat- t ice gauge t heo ry on o t h e r la t t ices than the p r imi t ive (hyper )cubic one [1-3]. T h e r ea sons are bo th t heo re t i ca l and prac t ica l , and re fe r essent ia l ly to the poss ib le inex is tence of uni- versa l i ty and to the poss ib i l i ty of a la t t ice- d e p e n d e n t r egu la r i ty of the c o n t i n u u m limit , respec t ive ly .

T h e in te res t in keep ing as much as poss ib le of the c o n t i n u u m s p a c e - t i m e s y m m e t r y l e a d s in a na tu ra l way to those la t t ices which admi t as s y m m e t r y g roups max ima l g roups (max imal a r i t hme t i c ho lohed r i e s ) [4-6] . A s far as one is no t fo rced to w e a k e n this max ima l s y m m e t r y r e q u i r e m e n t one has t h e r e b y dras t ica l ly r e d u c e d the n u m b e r of in te res t ing groups : to 2 in two d imens ions , to 4 in t h ree d imens ions , and to 9 in four d imens ions [5,6]. O n the o t h e r h a n d the g e o m e t r i c s t ruc ture r e l a t ed to the la t t ice group , which en te r s in to play, d e p e n d s very much on the t heo ry one inves t iga tes . A pure gauge theory has as its unde r ly ing s t ruc ture a net (of l inks) c o m p a t i b l e with the s y m m e t r y g roup [1,2]. If one adds f e rmions in a " n a i v e " m a n n e r , the r e l evan t s t ruc ture r e m a i n s the net . G e o m e t r i c f e rmions [7-9] r equ i r e in a la t t ice t heo ry [10-13] in d d imens ions c o m p l e t e d - d i m e n s i o n a l s truc- tures: a ce l lu lar pa r t i t i on of the euc l idean space

0 031-9163/83/0000-0000/$03.00 © 1983 N o r t h - H o l l a n d

(E d) c o m p a t i b l e with the s y m m e t r y group. To any b o u n d a r y e l e m e n t (corner , edge, v o l u m e . . . . ) and to the in te r io r of the cell i tself the re c o r r e s p o n d s one deg ree of f r e e d o m of a geome t r i c la t t ice fe rmion . T h e n u m b e r of deg rees of f r e e d o m N ( G ) of g e o m e t r i c la t t ice fe rmions , given a s y m m e t r y g roup G, is thus

d

N ( G ) = ~', C, , (1) i=0

where Ci, i = 1 . . . . . d, is the n u m b e r of (homology) chains of d imens ion i [ / -faces: cor- ners (i = 0), edges (i = 1) . . . . . the cell i tself (i = d)], which are not equ iva l en t u n d e r la t t ice t rans la t ion . This n u m b e r may be inc reased a rb i t ra r i ly , bu t it has a min ima l va lue Nm(G). T h e na tu r a l c a nd ida t e s as rea l i ze rs of these m i n i m a are the Di r i ch le t (or V o r o n o i ) cells [14] ,x. They are no t the only cand ida t e s : the ( topologica l ly) dua l s t ruc tures a re equa l ly well sui ted.

I t is p r e s u m a b l y so tha t the Di r i ch le t cells of the ma x ima l g roups d e t e r m i n e the min ima l n u m b e r Nm(G) of deg rees of f r e e d o m for geome t r i c f e rmions even on a la t t ice of a rb i t r a ry

,1 In solid state physics they are called Wigner-Seitz cells or (first) Brillouin zones, depending on whether one con- siders them in coordinate or momentum space.

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Volume 131B, number 1,2,3 PHYSICS LETTERS 10 November 1983

Table 1 Combinatorial description of the Dirichlet cells (and of their duals) of maximal arithmetic holohedries in 2, 3, and 4 dimensions. The four-dimensional lattice terminology is according to ref. [4]. The self-reciprocal lattices are marked by (+), pairs or reciprocal lattices always by a common sign. Selfduality is abbreviated as s.d.

Dimension Maximal arithmetic Co Cl C2 C3 C4 Nm(G) Dual structure Reciprocity holohedry (Lattice)

2 square 1 2 1 4 1 square (s.d.) + hexagonal 2 3 1 6 2 triangles +

(simplexes)

cubic, primitive 1 3 3 1 8 1 cube (s.d.) + hexagonal 2 5 4 1 12 2 triangular +

prisms cubic, 3 8 6 1 18 1 octahedron =

face-centered (F) +2 tetrahedra cubic, 6 12 7 1 26 6 tetrahedra =

body-centered (I) (simplexes)

hypercubic, primitive 1 4 6 4 1 16 1 hypercube + (s.d.)

hexagonal-tetragonal, 2 7 9 5 1 24 + primitive

cubic orthogonal, 3 11 14 7 1 36 x F(2,3,4)-centered

diisohexagonal 4 12 13 6 1 36 + orthogonal, primitive

icosahedral, primitive 4 15 20 10 1 50 / / cubic orthogonal, 6 18 19 8 1 52 x

I(2,3,4)-centered hypercubic, Z- centered 3 24 32 12 1 72 + diisohexagonal 20 54 48 15 1 138 +

orthogonal, RR2-centered icosahedral, 24 60 50 15 1 150 24 simplexes / /

SN-centered

symmetry group G by the formula

Nm(G) = rain Nm(GM), G C GM, (2) Gm

i.e. by the smallest of the minimal numbers Nm(GM)'of those maximal groups GM, to which the group G is a subgroup. When operating with nonmaximal groups it should thereby be kept in mind that there could be several Dirich- let cells for such a group. As example we quote the holohedry of the face-centered (or, equivalently, body-centered) tetragonal lattice. Out of the total number of 5 three-dimensional Dirichlet cells [15, 16] .2, there are 3 cells com- patible with the group: the (deformed) cells of

,2 For d = 2 their number is 2, and for d = 4 it is 52 [15].

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the face-centered (Nm = 18) and body-centered (Nm = 26) cubic lattices, to whose holohedries this holohedry is a subgroup, and in addition there is a cell (with N = 20) which does not correspond to any maximal holohedry. The minimal number of degrees of f reedom of geometric fermions (in the example = 18) can thus essentially not be reduced by lowering the symmetry; it can be changed by suitably choos- ing among the maximal symmetry groups.

In table 1 we give for the Dirichlet cells of the maximal arithmetic holohedries in d = 2 . . . . . 4 dimensions the numbers Nm, together with the numbers C~ of (equivalence classes under the group of lattice translations of) i- faces, i = 0 . . . . . d. The relation

Volume 131B, number 1,2,3 PHYSICS L E T F E R S 10 November 1983

d (-1)~C~ = 0 (3)

i=0

is a consequence of the toroidal topology of the cells, when considered f rom the point of view of our problem.

To the cell structures of table 1 there exist dual structures, which correspond to the change Ci ~ C~_~. They represent, as also (partly) shown in table 1, symmetry-adapted minimal divisions of the primitive parallelotopes (elementary unit cells) of the lattices into polytopes.

In each dimension there is a single selfdual (simple hypercubic) cell structure (cubic ho- mology), which realizes the smallest number 2 6 of degrees of f reedom. The dual structure to a Dirichlet cell of largest Nm [14],

d Nm = ~ Ci, (4)

i=0

C ~ - , = ~ ( - l ) i - k ( i k ) ( l + k ) a , k=O

represents the minimal simplicial decomposit ion of the e lementary unit cell (simplicial homology).

Although the minimal number of degrees of f reedom of geometric fermions is lattice dependent , the degrees of f reedom exceeding 2 d have to disappear, for free fermions, with vanishing lattice constant on any lattice. This follows f rom the existence and uniqueness of the Laplace-Bel t rami opera tor on a manifold (homology equivalence). These degrees of f reedom will affect the spectral multiplicity of the lattice opera tor at regions which escape to infinity when the lattice constant tends to zero. One may formulate this as the fact that the geometric formalism has built in it a Wilson mechanism, which enhances the masses of the additional degrees of f reedom (to infinity) with decreasing lattice constant. The manner in which this mechanism works in d = 2 dimen- sions is illustrated by the spectral analysis of ref. [17] ,3. What happens for interacting fermions can, of course, not be decided in this simple manner.

,3 The analogous two-dimensional example for naive fer- mions has been discussed in ref. [18].

We want to comment now on the relation, for the maximal symmetries, between the minimal number of degrees of f reedom of geometric lattice fermions and the minimal number of naive lattice fermions. These naive fermions are located on the minimal set of cri- tical points of the symmetric vector fields on the Dirichlet cell of the reciprocal lattice [19], i.e. on centers of i - facesof the Dirichlet cell in mo- mentum space (the first Brillouin zone). The degrees of f reedom of geometric fermions, on the other hand, may be considered as located on the centers of the / - faces of the Dirichlet cell of the lattice in coordinate space (the Wigner-Seitz ceil). So they should be expected to be equal in number only when these two cells coincide.

In table 1 we have marked the self-reciprocal and pairs of reciprocal cells. In two dimensions both cells are self-reciprocal, whereas in three dimensions only those of the primitive lattices are so; the cells of the two centered cubic lat- tices are reciprocal to each other. On the body- centered cubic lattice there have to be at least 26 degrees of f reedom for geometric fermions. The Brillouin zone of this lattice has 18 /-faces (i = 0 . . . . . 3), but of the 18 centers of these faces only 10 (= Co + C2 + 6?3) have to be critical points [16,19] (locations of naive fermions). On the face-centered cubic lattice one has at least 18 degrees of f reedom for geometric fermions. Its Brillouin zone has 26 / - faces (i = 0 . . . . . 3), but again only part of them, Co + C2 + 6?3 = 14 [16,19], have to bear in their centers naive fer- mions.

The minimal number of naive fermions on self-reciprocal lattices is at most equal to the minimal number of degrees of f reedom of geometric fermions. In two and three dimen- sions (and presumably also in four) these num- bers are, in fact, equal. For lattices which are not self-reciprocal for d = 3 the minimal number of naive fermions is smaller than that of the minimal degrees of f reedom of geometric fer- mions. With respect to d = 4 it should be emphasized that the centered hypercubic lattice is also self-reciprocal, and so one should expect that the number of 72 naive fermions [3] is the minimal one for this lattice. The precise rela-

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Volume 131B, number 1,2,3 PHYSICS LETI'ERS 10 November 1983

t ions for all d = 4 groups can be es t ab l i shed [19] by the c o m p u t a t i o n of the i so la ted orb i t s of the groups on the Di r ich le t cells.

T h e min ima l n u m b e r of degrees of freedom of naive f e rmions is, however , in all d ~< 3 cases a pr ior i la rger than tha t of geome t r i c fe rmions . It has been r e d u c e d th rough su i tab le p ro j ec t i on [20] for the p r imi t ive cubic la t t ice by the r ight factor , This p ro j ec t i on amoun t s to an ind iv idua l d ispers ion , in a cer ta in manne r , of the degrees of f r e e d o m on la t t ice si tes closely g r o u p e d toge ther .

T h e sp read ing of the deg rees of f r e e d o m in a b o u n d e d reg ion ove r severa l la t t ice d i s tances in c o o r d i n a t e space is p rec ise ly the p h e n o m e n o n for which the geome t r i c a p p r o a c h p rov ides a universa l p resc r ip t ion . Name ly , the deg rees of f r e e d o m of geome t r i c f e rmions can be con- s idered , as a l r eady men t ioned , to be l oca t ed at the cen te r s of t h e / - f a c e s of the Dir ich le t cell. N o n e of these points , except the cen te r of the cell itself, is a la t t ice point . But a long the rays of the vec tors jo in ing the cen te r of the cell with the cen te rs of the t r ans la t iona l ly inequ iva len t cen te r s of / -faces (i = 0 . . . . . d - 1) one can pick up a set of la t t ice si tes mos t c losely s i tua ted to the cen te r of the cell. This set, which is un ique up to equ iva lence , can be v iewed as the la t t ice suppor t for the min ima l degrees of f r e e d o m of geome t r i c fe rmions . T h e na ive f e rmions can be cons ide red as loca ted on prec ise ly the same set and in this sense the geome t r i c and na ive des- c r ip t ions of f e rmions on se l f - rec iproca l la t t ices a re equ iva len t . On lat t ices which are not self- r ec ip roca l the re is no equ iva l ence b e t w e e n the two desc r ip t ions of fe rmions ; the geome t r i c desc r ip t ion may then be v iewed as the na tu ra l ex tens ion of the Sussk ind p ro j ec t i on [20] to this k ind of lat t ices.

The c o m p a r i s o n we have d iscussed refers only to the max ima l groups (maximal a r i thmet i c

ho lohedr ies ) . The survival of equ iva l ence for the i r subgroups d e p e n d s on the min ima l sets of cri t ical po in t s of these subgroups on the Dir ich- let cells [19].

A de ta i l ed account of the a rgumen t s will be given in a fo r thcoming pape r .

I wou ld l ike to thank V. R i t t e n b e r g for many s t imula t ing ques t ions and commen t s .

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