Angular momentum on a lattice

Volume 114B, number 2,3 PHYSICS LETTERS 22 July 1982

ANGULAR MOMENTUM ON A LATTICE

R.C. JOHNSON CERN, Geneva, Switzerland 1

Received 1 March 1982 Revised manuscript received 28 March 1982

Reduction of SU 2 with respect to the double cover of the octahedral group shows how spin is realized on a cubic lattice. Only spins 0, 1/2, 1 and 3/2 are described by single octahedral irreducible representations; higher spins are described by two or more. The reduction procedure is accompanied by useful methods for constructing octahedral wave functions. Remarks on angular momentum assignments in lattice dynamical calculations are illustrated by a simple soluble model.

1. Introduction. The group of rotational symme- tries of a cube is the "angular momentum" group for the cubic lattice. It is the well-known [1] crystallo- graphic point group O (octahedral group) and is o f course a finite subgroup of the continuum rotation

group SO 3 . As is familiar, SO 3 has the infinite sequence of vec-

tor irreducible representations (IRs) labelled by integer angular momen tum] = 0, 1, 2 .... plus the inter- leaving sequence of ray IRs for half odd-integer angular momen tum] = 1/2, 3/2, 5/2 . . . . . Together these are the vector IRs o f its double cover SU 2.

However, O has only five vector IRs and a further three ray IRs that together correspond to the eight vector IRs o f 20, its double cover.

That is, only eight different possibilities exist for rotational classification of states on a cubic lattice. So the question arises: exactly how do these correspond to angular momentum in the continuum?

The question is interesting because of increasingly ambitious QCD calculations of hadron states, includ- ing fermions, on a simple cubic lattice [2].

The answer is to some extent ( for] = 0, 1 ,2 at least) well known [2,3] and more can be extracted from the solid-state literature [4]. The purpose of this note is to summarize the complete answer, includ- ing fermions.

i Permanent address from 1 April 1982: Durham University, UK.

The methods used are quite standard in the crystal- lographic context [1,5], and give straightforward pro- cedures for reduction of arbitrary representations of O and 20, and the construction of wave functions with octahedral symmetry.

Clearly, SU 2 representations map many-to-one (oo to 8) into the IRs of 20, and so the inversion is singular. That is, there is ambiguity in interpreting the spin content of lattice states - at least, without additional dynamical assumptions. Remarks on this issue are illustrated by reference to a group theoretically soluble model - the isotropic simple harmonic oscillator.

2. Groups and representations. The octahedral group O is isomorphic to the full tetrahedral group T d (rotations and reflections of a regular tetrahedron), and to the permutation group of four objects, S 4, (which can be thought of as the tetrahedron's vertices or the cube's body diagonals).

The 24 elements of O, enumerated in their five conjugacy classes, are as follows - where reference is to cartesian coordinates x, y, z with origin at the cube's center and axes parallel to its edges:

I: identity. 3C2: n rotations about 3 coordinate axes. 8C 3 : -+(2n/3) rotations about four body diagonals,

e . g . x = y = z . 6C4:-+Ir/2 rotations about 3 coordinate axes. 6C~: lr rotations about axes parallel to 6 face diag-

onals, e.g. x = y , z = 0.

0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland 147


Notation nC m means a class of n elements of order m - i.e. each is an rnth root of the identity.

Recalling [5] that for finite groups the number of (vector) IRs equals the number of conjugacy classes, there are five IRs of O. They are usually [4] called A1, A2, E, T1, T2, with respective dimensions 1, 1, 2, 3, 3. (Recall also that the sum of squares of dimensions of IRs equals the number of group elements.) Of these, A 1 is the identity representation and T 1 rotates the geometrical vector (x, y , z). A change of sign of matrices in both classes C 4 and C~ connects A 2 to A 1 and T 2 to T 1. Slater [1, p. 358], gives matrix elements, although with misprints. It is convenient to denote these five IRs collectively by qY, say.

The group 20, the double cover of O, is constructed by adding a negative identity J for -+ 27r rotations. Then from the C 3 and C 4 conjugacy classes of O come two new classes of elements of 20, of order six and eight, respectively. Thus, with J itself, the 48 elements of 20 divide into 8 conjugacy classes.

Of the consequent 8 IRs of 20, five are given by cl~, with the extra 24 matrices in each IR being the negative of the 24 O matrices. The 3 further IRs o f 2 0

give ray representations of O - call them collectively c~.

The dimensions of c~ must be even, because for in- stance rotations of rr about x and y axes square to give minus the identity, and anticommute with each other. It is the standard reasoning [6] for Dirac matrices. Only dimensions 2, 2, 4 fit the dimension sum role (al- ready half saturated by qY), and these IRs are called G1, G2, H, respectively. Matrices of G 1 are obtained from Pauli o matrices, since a rotation through angle 0 about unit vector n is represented by

R(n, 0) = I c o s ½ 0 - i , - n s in{0,

where I is the 2 × 2 unit matrix. Then G 2 is related to G 1 like A 2 to A 1 and T 2 to T 1 . As will appear, matrices of H are given by ] = 3/2 SU 2 matrices. It should be noted that the matrices for ray representations of O given by Hurley [7] are wrong - they perpetuate an error in the original work of DiSring [8]. (In these authors' notation dc = c2da, not dc = c2d.)

3. Lattice spin and wave functions. Table 1 gives characters o f c)y andC~, treating ely as vector IRs of O, and c~ as vector IRs of 20. It is clear that vector representations of SO 3 (/" = 0, 1, 2 .... ) reduce to -qY and the

Table 1 Irreducible characters of O and 20 . Values of 0 are included to calculate traces of corresponding SU2 matrices from eq. (1).

(a) vector IRs ely of O

I 3C 2 8C 3 6C 4 6C~

A 1 1 1 1 1 1 A 2 1 1 1 - 1 - 1 E 2 2 - 1 0 0 T 1 3 - 1 0 1 - 1 T 2 3 - 1 0 - 1 1

O 2rr rr 2rr/3 lr/2 lr

(b)addi t~nalvec tor lRsof20 - r a y l R s o f O , CR

I J 6C4 8C3 8C 6 6Ca 6C~ 12C~

G1 2 -2 0 -1 1 -x/~ x/~ 0 c~ 2 - 2 o -1 1 ~ - ~ o H 4 - 4 0 1 - 1 0 0 0

0 47r 21r lr 47r/3 27r/3 37r/2 7r/2 n

other SU 2 representations ( j = 1/2, 3/2, 5/2 .... ) reduce toC~. The reduction is carried out using orthogo- nality properties of characters [9], as follows.

The traces of SU 2 matrices for spin] and rotation angle 0 are given by

X~ j) = sin[(j + 1/2)0] /s in[0/2] , ( I )

so the characters of the generally reducible representations of O and 20 provided by these ( 2 / + 1) 2 matrices are easily calculated; [the various angles 0 are given in table 1 and limits must be taken in eq. (1) if necessary].

Then the appropriate projection formula for the tr) multiplicity m/: of an octahedral IR r in spinj reads

-- 1 23 n v(Ov(" (2) g k k'~k Ak '

where there are n/c elements in conjugacy class k with irreducible character ×(k O, and g = ~,knk = 24 (for ),

or 48 (for c~). Also xf) stands for ×~J) from eq. (1), with

angles O k in table i. Note that all characters are real. As far as group theory goes, this is the same as the calcula- tion of multiplet splitting in a crystal field [ 101.

Table 2 gives consequent results up to j = 6, and continuation is routine. Evidently only angular momenta j = 0, 1/2, 1 ,3/2 correspond to single IRs of

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Table 2 Reduction of SU 2 to 20, up toj = 6.

0 A 1 1/2 G1 1 T 1 3/2 H 2 E + T 2 5/2 G 2 + H 3 A 2 + T 1 + T 2 7/2 G1 + G2 + H 4 AI +E+T1 +T 2 9/2 G1 + 2H 5 E + 2T 1 + T2

11/2 G 1 +G 2 +2H 6 A I + A 2 + E + T 1 +2T 2

both SU 2 and 20. Higher spinsj ~> 2 are described by two or more different IRs of the octahedral group. No reason appears why reduced rotational symmetry alone could lead to species multiplication of lattice fermions [111.

Reduction of SU 2 harmonics to basis states o f octahedral IRs is carried out with similar ease.

Define the operator [9]

n p(r) x(r)(R)R, (3)

g R

where the sum runs over all g group rotations R, and ×(r)(R) is the (real) character of R ' s matrix representa- tive in IR r of dimension n r. Then in fact p(r)ff trans- forms like a basis vector of IR r, where ~b is any wave function such that R ~ makes sense. Different choices of ~ build up a complete basis for r, (but there is ambiguity if ~ Contains r more than once).

An alternative projection operator is given by, e.g., Slater [1] [his eq. (A12.42)], using matrix elements (i.e. no trace taken). It decomposes - effective- ly partial-wave projects - a single ¢s into up to n r com- ponents.

Either projector can be used to construct wave functions of definite octahedral symmetry, not just from SU 2 harmonics, but also from harmonic oscillator wave functions, or from Wilson loop operators [2, 11], or whatever. Eq. (3) is most useful for constructing IRs E and H, since they have two or more classes with zero character, and Slater's form is very convenient for IRs T 1 and T2, which have many zero matrix elements.

4. lnverse problem. A specific question prompted by lattice QCD calculations of the hadron spectrum [2] is: what is the spin content o f a lattice energy eigenstate transforming as a single IR of O?

That is, what is the inverse of the correspondence specified by eq. (2) and table 2?

The answer is: A 1 ~ ( j = 0 , 4 , 6 , 8 .... ), A2 ~ ( j = 3, 6, 7, 9 .... ), E - + ( ] = 2, 4, 5, 6 .... ), T 1 ~ (] = 1, 3, 4, 5 (twice) .... ), T 2 - * ( ] = 2 , 3 , 4 , 5 .... ), G 1 ~ ( j = 1/2, 7/2, 9/2, 11/2 .... ), G 2 ~ (j = 5/2, 7/2, 11/2, 13/2 (twice) .... ), H ~ ( j = 3/2, 5/2, 7/2, 9/2 (twice) .... ).

Ascribing a defmite spin to a single lattice state is therefore not possible without further dynamical in- formation or assumptions.

However, presumably few would quarrel with bosonic and fermionic ground states in representations A 1 and G 1 being identified as spins 0 and 1/2, respectively - especially since the next contributing angular momenta are j = 4, and j = 7/2. Perhaps also it is reasonable to identify a low-mass T 1 state asj = 1 - but if in the continuum limit hitherto distant A 2 and T 2 levels moved to coalesce with it, t hen j = 3 would be there, either instead, or as well. Similarly the inter- pretation of H asf = 3/2 alone is uncertain if G 2 could move in to make] = 5/2.

Is an E state spin 2? Maybe, if T 2 is nearby; and yes, if they (and they alone) coincide in the continuum limit. But without following both to the limit the massM is not well defined. The best that can be done is to take

M(j = 2) = some average of (M(E), M(T2)),

where the averaging procedure to be used is a dynamical question (and where the absence o f j = 4, 5, etc., is not guaranteed in general). For higher spins, where more representations of O contribute, mass predictions are correspondingly harder, even if all the IRs needed can be identified. Ishikawa et al. [2] propose a dynamically motivated averaging procedure in their glueball calculations.

5. Soluble model Illustration of the possible am- biguities in angular momentum classification of a lattice system can be given in a soluble non-relativistic

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Volume l14B, number 2,3 PHYSICS LETTERS 22 July 1982

quantum mechanical model - the three-dimensional isotropic simple harmonic oscillator on a cubic spatial lattice. The word "soluble" here means that it is possible to enumerate and classify group theoretically the lattice system's energy levels. There is no claim to have any lattice analogue of the explicit formula for them that is familiar [12] in the continuum!

The point is that the hamiltonian ~ is just a sum of three similar and mutually commuting x, y, z pieces of the form p2 + x 2. So energy levels are sums of exci- tations in each of the three directions, and wave functions are simple products.

On an n 3 lattice each of the three pieces o f ~ is an n X n matrix in lattice-site space (irrespective of the choice o f approximation for momentum operator p), and so gives n energy levels. However, because of the restriction to a finite lattice, these are by no means equally spaced, as in the continuum.

For example [13], with the SLAC derivative [14] for p and periodic boundary conditions, when the lattice is big enough to reasonably hold the wave functions and freely-spaced enough to resolve their spatial variation, the lowest few levels approximate quite well their continuum counter-parts, and then there is a transition to a strong-coupling regime of parity dou- blets, with the highest levels heavily distorted by the boundary conditions.

Therefore in three dimensions the lattice oscillator spectrum is a jumbled sequence of levels of degenera- cies 1, 3 and 6, corresponding to the (non-accidental) degeneracy o f E a + E b + E c = fixed, with a, b, c (= 1, 2 ..... n independently) either all the same; or with two the same, or all different. Both six-fold and three-fold levels have four different possible parity con- tents (in an obvious notation +++, ++- , ~ , ) and the non-degenerate levels are either +++ or - - -

Characters of the levels as representations of O are easily calculated, and then eq. (2) and table 1 give their reduction to c~. Mixed parity six-fold levels reduce to T 1 + T2, and single-parity levels to A 1 + A 2 + 2E. The four parity types of three-fold levels listed above reduce to A 1 + E, T1, T2, A 2 + E, respectively, and the non-degenerate levels are A 1 (+++) or A 2 ( - - - ) . Eq. (3) can be used to obtain wave functions if desired.

Table 3 gives the oscillator spectrum for a case n = 3. Even on such a small lattice the spectrum is rich - it has 10 levels, corresponding to the lowest 7 of

Table 3 Spectrum of an isotropic simple harmonic oscillator on a 33 lattice, in order of increasing energy. Levels El, 2,3 of the three one-dimensional subsystems have E 2 - E 1 > E 3 - E 2 , with respective parities +, +, - . Exact numerical values of EI,2,3 are irrelevant.

Quan tum niJmbers a) Degeneracy Parity O content

111 1 +++ Al 112 3 +++ A 1 + E 113 3 + + - TI 122 3 +++ A1 + E 123 6 ++- T1 + T2 133 3 + - - T 2 222 1 +++ A l 223 3 ++- T 1 233 3 + - - T 2 333 1 A 2

a) Here abc means that the level is formed f rom one-dimensional levels with energies Ea, E b and E c.

the continuum system, which has equal spacing. How- ever, there is distortion by (periodic) boundary conditions so that successive contributing levels in the one- dimensional subsystems have parities +, +, - and their wave functions have 0, 2, 1 nodes, respectively.

Consequently only the ground state (111) is imme- diately and confidently identified asj = 0.

The next level (112) with IRs A 1 + E is part of the continuum j = 0, 2 tower (second excited level) but there are two different T 2 possibilities (133,233) with the correct parity to partner it. In fact the nearer (133) is the right one, but even if correctly identified the splitting is large, and without going to the continuum limit the energy o f / = 2 is not defined.

The third level (113) is the lowest IR T1, and so reasonably (and correctly) identified as] = 1 - although on the lattice it is forced above part of the j = 0, 2 multiplet.

Fourth and fifth levels (122, 123) are parts o f ] = 0, 2, 4 a n d / = 1, 3 towers, respectively, with continuum ordering reversed. Distant levels 233 and 333 are in fact respective partners, but because of the small size of the lattice the complete multiplets are missing. These, and other levels more so, are just shadows of their continuum counterparts, without knowledge of which angular momentum assignments would be im- possible.

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6. Conclusion. To summarize, we have shown how to put spin onto a cubic latt ice using a quite general and straightforward method, which goes alongside similarly simple methods for constructing wave functions of definite irreducible cubic symmetry. Applicat ion to other shapes of regular spatial latt ice merely involves different point groups.

Discretisation of space reduces the infinity of IRs of SU 2 into a small number of lattice-group IRs. A simple example illustrates the point that the inverse pro- cess is singular in principle and possibly highly ambigu- ous in practice. Great care is needed in making predictions about masses and spins of latt ice states, especially beyond the lowest levels and for ] > 3/2. To be sure of higher spin assignments and mass predict ions it seems necessary to follow all relevant lattice IRs simul- taneously to the continuum limit. Otherwise, excited meson and glueball predictions must be uncertain, and when baryons are calculated perhaps a nucleon mass may be credible but parameters of N* states, etc., could be wildly inaccurate.

Note added. The group theory calculations in this paper are the same as the calculations of mult iplet splitting in crystal fields that are described in refs.

[15,16].

References

[1] J.C. Slater, Quantum theory of molecules and solids, VoL ! (McGraw-Hill, New York, 1963) Appendix 12.

[2] A partial list includes: H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792; E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795; H. Hamber, E. Marinari, G. Parisi and C. Rebbi, Phys. Lett. 108B (1982) 314; D. Weingarten, Phys. Lett. 109B (1982) 57;

A. Hasenfratz, Z. Kunszt, P. Hasenfratz and C.B. Lang, CERN preprint TH.3220 (Dec. 1981); B. Berg, Phys. Lett. 97B (1980) 401; G. Bhanot and C. Rebbi, Nucl. Phys. B180 [FS2] (1981)469; B. Berg, A. Billoire and C. Rebbi, Ann. Phys. (NY), to be published. B. Berg and A. Billoire, CERN preprint TH.3230 (Jan. 1982); M. Falcioni et al., Phys. Lett. l l0B (1982) 295; K. Ishikawa, M. Teper and G. Schierholz, Phys. Lett. l l0B (1982) 399.

[3] J.M. Drouffe and C. Itzykson, Phys. Rep. 38C (1978) 133.

[4] E:g. S.L. Altmann and A.P. Cracknell, Rev. Mod. Phys. 19.

[5] W. Miller, Symmetry groups and their applications (Academic Press, New York, 1972) Chs. 2 and 3.

[6] E.g.J.D. Bjorken and S.D. Drell, Relativistic quantum mechanics (McGraw-Hill, New York, 1964) p. 8.

[7] A.C. Hurley, Phil. Trans. R. Soc. London, Set. A 260 (1966) 1.

[8] W. D6ring, Z. Naturforsch. 14a (1959) 343. [9] W. Miller, Symmetry groups and their applications

(Academic Press, New York, 1972) section 3.5. [ 10] E.g.J.C. Slater, Quantum theory of molecules and solids,

Vol. 1 (McGraw Hill, New York, 1963) section A12-8. [11] See, e.g.P. Hasenfratz, Topics in non-perturbative QCD,

talk 1981 Bonn Lepton-photon Symp. (August 1981), CERN preprint TH.3187 (October 1981), and references therein.

[12] See, e.g.L.I. Schfff, Quantum mechanics, 3rd Ed. (McGraw-Hill, New York, 1968) sections 13, 25 and 30.

[13] R.C. Johnson, in preparation. [14] S.D. Drell, M. Weinstein and S. Yankielowicz, Phys. Rev.

D14 (1976)487, 1627; J.M. Rabin, Phys. Rev. D24 (1981) 3218.

[15] M. Hamermesh, Group theory and its application to physical problems (Addison-Wesley, London, 1962) sections 9.4 and 9.7.

[16] M. Tinkham, Group theory and quantum mechanics (McGraw Hill, New York and London, 1964) sections 4.5-4.7.

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