Enlarged Symmetry Groups of Finite-Size Clusters with Periodic Boundary Conditions

Any system that approximates an infinite lattice by a family of finite clusters (with periodic boundary conditions) passes through an intermediate region with enlarged (hidden) symmetry as the system size is increased. The hidden symmetry allows for extra degeneracies and level crossings and has application to exact-diagonalization studies, Monte Carlo simulations, lattice gauge theories, and renormalization group calculations. AMS subject classifications (1991). 82D25, 20H15, 20C30. The periodic crystal approximation [1] is the fundamental approximation for studying bulk properties of solid-state systems. It has been used quite successfully in band-structure calculations [2], Monte Carlo simulations [3], and the smallcluster approach to the many-body problem [4]. In the periodic crystal approximation a crystal of M sites is modeled by a lattice of M sites with periodic boundary conditions (PBC). Bloch's theorem [5] then labels the quantum-mechanical wavefunctions by one of M wavevectors in the Brillouin zone [6]. In principle, the thermodynamic limit (M ~ ~ ) is then taken which replaces the finite grid in reciprocal space by a continuum that spans the Brillouin zone (or equivalently replaces the finite cluster by an infinite lattice in real space). In practice, the number of lattice sites is chosen to be as large as possible (M = finite), and the solution of the quantum-mechanical problem corresponds to a finite sampling in reciprocal space. It should be emphasized that every calculation that samples at only a finite number of points in reciprocal space corresponds to a finite-cluster with periodic boundary conditions in real space. Realistic models of physical systems (that incorporate many-body interactions on the same footing as single-particle interactions) can be analyzed for only the smallest systems. Recent examples include calculations of the photoemission spectra [7] of nickel (M = 4), structural properties [8] of diamond and silicon (M = 8), and x-ray absorption spectra [9] in high-temperature superconducting oxides (M = 10). It is not known, in general, how large a finite system must be to capture the relevant physics of a real system (in the thermodynamic limit). Group theory can be used to address this question by identifying when extra symmetries of finite clusters (with 278 J. K. FREERICKS AND L. M. FALICOV periodic boundary conditions) exist and by analyzing the representation theory of these enlarged groups. Group theory is a rigorous and model-independent tool that determines when particular matrix elements are required to be, e.g., identical or zero. Such analysis leads to two possible finite-size effects of the hidden symmetry when compared to the standard analysis based upon the space group: energy levels can 'stick together' and/or 'violate' the no-crossing rule. A finite cluster with PBCs can be viewed as a mapping of an infinite lattice onto M equivalence classes each class corresponding to a different site of the cluster [10] (for example, any bipartite [11] lattice of translationally equivalent sublattices can be represented by a two-site cluster with the ~ sublattice corresponding to equivalence class one and the fl sublattice corresponding to equivalence class two). The lattice contains only M inequivalent translations since any translation that does not change the equivalence classes of the sites is made identical to the null translation (for example, the two-site cluster has two translations; the first corresponds to the null translation and carries sublattice ~--*a and sublattice fi ~f l ; while the second corresponds to the translation from one site to the other and carries sublattice a ~ ~ fl). The neighbor structure of the lattice is determined by the successive nearest-neighbor shell (on-site, first-nearest-neighbor, second-nearestneighbor, etc.) that exhaust all M equivalence classes (the two-site cluster contains only on-site and, normally, first-nearest-neighbor shells). Note that each neighbor shell may contain members of an equivalence class more than once (if the bipartite lattice has Z nearest neighbors then the nearest neighbors on site one are Z atoms of equivalence class two). The space group is finite and consists of Mh elements, where h is the number of elements in the point group (the largest value for h is 48 for cubic lattices; it is 8 for square lattices). In the thermodynamic limit (M ~ ~ ) the complete symmetry group of the lattice is the (infinite) space group (with Mh elements), which is composed of all translations, rotations, and reflections that (rigidly) map the infinite lattice onto itself and preserve its neighbor structure. In the case of a finite cluster, the complete symmetry group is a subgroup of SM, the permutation group of the M cluster sites, and is called the cluster-permutation group. The cluster-permutation group, which contains all operations that leave the Hamiltonian invariant, may (A) be a proper subgroup of the (finite) space group (i.e. it has fewer elements than the space group), (B) contain operations that are not elements of the (finite) space group, or (C) be identical to the (finite) space group. These three regimes are called, respectively, (A) the self-contained-cluster regime, (B) the high-symmetry regime, and (C) the lattice regime). Note that the whole (finite) space group need not be a subgroup of the cluster-permutation group in the high-symmetry regime (although it usually is). A self-contained cluster (A) is a cluster essentially identical to an isolated, box-boundary-conditions cluster. The addition of PBC adds no new connections between lattice sites, but the neighbor structure of the cluster with PBCs may contain multiples of the neighbor structure of the isolated cluster (thereby renorENLARGED SYMMETRY GROUPS OF FINITE-SIZE CLUSTERS 279 malizing parameters in the Hamiltonian in going from one to the other). In this case, the cluster-permutation group is identical to the symmetry group of the same isolated cluster. This symmetry group is a point group, not necessarily the full point group of the lattice; it is a proper subgroup of the space group. This isomorphism was first observed [10, 12] in the 2 x 2 square (sq) and the four-site tetrahedral (face-centered cubic, fcc) clusters (M = 4) and in the 2 x 2 • 2 simple-cubic (sc) cluster (M = 8). In the self-contained-cluster regime, the cluster-permutation group is a proper subgroup of the space group, because some space-group operations are redundant (i.e., there is a group homomorphism between the space group and the cluster permutation group with a nontrivial kernel). The redundancy implies that only a sub-set of the irreducible representations of the space group (those that represent the elements of the kernel by the unit matrix) are accessible to the solutions of the Hamiltonian. This process of rigorously eliminating irreducible representations as acceptable representations is well known. It occurs, for example, in systems that possess inversion symmetry: if the basis functions are inversion symmetric, then the system sustains only those representations that are even under inversion. For intermediate-size clusters there are additional permutation operations that leave the Hamiltonian invariant. They either (nonrigidly) map the lattice onto itself and preserve the entire neighbor structure of the lattice, or (for short-range-interaction Hamiltonians) they preserve only the first-nearest neighbor (1NN) structure of the lattice [ 13]. These hidden symmetry operations may expand the cluster-permutation group to a group that is (much) larger than the (finite) space group. The group theory for the cluster-permutation group in the high-symmetry regime (B) may be analyzed as follows. The set H of elements of the cluster-permutation group G that are elements of the space group forms a subgroup of the clusterpermutation group that, usually, is equal to the space group [14]. The group of translations T forms an Abelian invariant subgroup of H; the irreducible representations of H are all irreducible representations of the space group. When the full cluster-permutation group G is considered, the class structure of H is expanded and modified, in general, with classes of H combining together, and/or elements of G outside H uniting with elements in a class of H, to form the new class structure of the cluster-permutation group G. The classes that contain the set of translations typically contain elements that are not translations, so that the translation subgroup is no longer an int, ariant subgroup and representations of the cluster-permutation group cannot be constructed in the standard way [15]. Furthermore, every irreducible representation of H that has nonuniform characters for the set of classes of H that have combined to form one class of G must combine with other irreducible representations to form a higher-dimensional irreducible representation of the cluster-permutation group. This phenomenon can be interpreted as a sticking together of irreducible representations of the space group (i.e., the subgroup H) arising from the extra (hidden) symmetry of the cluster. The lattice-regime clusters appear for large enough M, assuming that the unit cell is chosen with enough symmetry. In this regime (C) the group properties are 280 J. K. FREERICKS AND L. M. FALICOV Table I. Number of symmetry elements in the space and the cluster-permutation groups (CPG) for arbitrary interactions on finite-size clusters with periodic boundary conditions of the simple, body-centered, and face-centered cubic lattices and of the two-dimensional square lattice. The symbols, A, B, and C denote the self-contained, high-symmetry, and lattice regimes, respectively. The cases with cluster sizes larger than 32 are all in the lattice regime (C). Cluster Space CPG Space CPG size group group cubic sc bcc fcc square sq 1 48 A 1 A 1 A 1 8 A 1 2 96 A 2 A 2 16 A 2 4 192 A 24 A 8 A 24 32 A 8 8 384 A 48 B 1152 B 384 64 B 128 16 768 B 12 288 B 4608 B 7 962 624 128 C 128 32 1536 C 1536 C 1536 C 1536 256 C 256 completely determined by the space group, and the irreducible representations are labeled by a wavevector in the Brillouin zone and (at symmetry planes, lines and points) by a subindex that determines the relevant irreducible representation [16] under rotations and reflections. The wavevector k labels the characters of the Abelian invariant subgroup of translations by determining the phase change exp(ik, z) for a translation z (Bloch's theorem). The transition from (A) self-contained cluster to (B) high-symmetry cluster, to (C) lattice is illustrated in Tables I and II for the simplest set of cubic (sc, bcc, and fcc) and square (sq) lattice clusters: the set whose number of sites is a power of two (M = 2~). These sets can all be constructed with maximum cubic or square symmetry, with the exception of the M -2 cluster for the fcc lattice. The tables record the sizes of the space group and the cluster-permutation group for the chosen set of clusters. Table I corresponds to arbitrary Hamiltonians; Table II, to Table II. Number of symmetry elements in the space and the cluster-permutation groups (CPG) for INN-only interactions on finite-size clusters with periodic boundary conditions of the simple, body-centered, and face-centered cubic lattices and of the two-dimensional square lattice. The symbols, A, B, and C denote the self-contained, high-symmetry, and lattice regimes, respectively. The cases with cluster sizes larger than 128 are all in the lattice regime (C). Cluster Space CPG Space CPG size group group cubic sc bcc fcc square sq 1 48 A 1 A 1 A 1 8 A 1 2 96 A 2 A 2 16 A 2 4 192 A 24 A 8 A 24 32 A 8 8 384 A 48 B 1152 B 384 64 B 1152 16 768 B 12 288 B 3 251 404 800 B 7 962 624 128 B 384 32 1536 B 13 824 B 6144 C 1536 256 C 256 64 3072 B 27648 C 3072 C 3072 512 C 512 128 6144 C 6144 C 6144 C 6144 1024 C 1024 E N L A R G E D S Y M M E T R Y G R O U P S OF FINITE-SIZE CLUSTERS 281 Hamiltonians with only INN-interactions. Notice that for (A) the number of elements in the cluster-permutation group ncp c is always smaller than in the space group ns; for (B) usually ncpo > ns, although it is possible to have ncpG = ns (see Table I, fee cluster of 8 sites); for (C) always ncpG = ns. The self-contained-cluster regime (A) corresponds to M ~< 8 [M ~<4] for the sc lattice [otherwise]. The high-symmetry regime (B) is present at intermediate values of M: for example, when the Hamiltonian contains only 1NN interactions the high-symmetry regime appears for 16~<M~<64 in the sc lattice; 8~<M~<32 in the bcc lattice; and 8 ~< M ~< 16 in thefcc and sq lattices (see Table II). The lattice regime (C) is entered for larger cluster sizes. The cluster-permutation group (in the high-symmetry regime) has been studied for some specific clusters [17 19]. As an example, the transition from a self-contained cluster to the lattice regime can be examined in more detail for the fcc lattice. The four-site tetrahedral fcc cluster is a self-contained cluster (see Figure 1). The nearest neighbors of each site of the tetrahedron are the other three sites. The imposition of PBCs produces an fcc lattice (see Figure la) in which each of the four interpenetrating sc sublattices of the fcc lattice is assigned to a different equivalence class. The twelve nearest neighbors of each site are now four atoms of each of the other three equivalence classes (see Figure I b). The only difference between the tetrahedral cluster with box boundary conditions (representing an isolated tetrahedron) and the tetrahedral cluster with periodic boundary conditions (representing an fcc lattice) is the 1NN interactions in the Hamiltonian are renormalized by a factor of four in the latter case. The space group is of order 192 (4 x 48) and has 20 irreducible representations: 10 with wavevector at the center of the Brillouin zone F {k = [0, 0, 0]}, and 10 with wavevector at the center X {k -[Tr, 0, 0], k = [0, re, 0], k = [0, 0, z~]} of the square