Sublattice coding algorithm and distributed memory parallelization for large-scale exact diagonalizations of quantum many-body systems

We present algorithmic improvements for fast and memory-efficient use of discrete spatial symmetries in exact diagonalization computations of quantum many-body systems. These techniques allow us to work flexibly in the reduced basis of symmetry-adapted wave functions. Moreover, a parallelization scheme for the Hamiltonian-vector multiplication in the Lanczos procedure for distributed memory machines avoiding load-balancing problems is proposed. We demonstrate that using these methods low-energy properties of systems of up to 50 spin-$1/2$ particles can be successfully determined.

Figure 1

Two sublattices coding of the spin state $|\mathbf{\sigma }\rangle$ on a six-site chain lattice and action of translational symmetries. The sites are enumerated such that sites 1–3 are on the blue (solid) sublattice $A$, 4–6 on the red (dashed) sublattice $B$. The representative state with this enumeration of sites is given by $|\tilde{\mathbf{\sigma }}\rangle =T_1|\mathbf{\sigma }\rangle =|\downarrow \downarrow \uparrow \uparrow \uparrow \downarrow \rangle$. Notice that the symmetries act on real space, and thus the transformation of the basis states also depends on the numbering of sites.

Figure 2

Sublattice orderings for several common lattices. The sublattices are distinguished by different colors (shadings). Left: Two sublattices ordering in a honeycomb lattice. The sublattices are stable with respect to all spatial symmetries. Middle: Three sublattices ordering on a kagome lattice. The sublattices are stable with respect to all spatial symmetries. Right: Three sublattices ordering on a square lattice. The sublattices are stable with respect to all translational symmetries, horizontal and vertical reflections, and ${180}^{\circ }$ rotations but not with respect to ${90}^{\circ }$ rotations or diagonal reflections.

Figure 3

Three sublattices coding of the spin state $|\mathbf{\sigma }\rangle$ on a six-site chain lattice and action of translation symmetries. The sites are enumerated such that sites 1 and 2 are on sublattice $A$ (blue, solid), 3 and 4 on $B$ (red, dashed), and 5 and 6 on $C$ (yellow, dotted). The representative state with this enumeration of sites is given by $|\tilde{\mathbf{\sigma }}\rangle =T_2|\mathbf{\sigma }\rangle =|\downarrow \downarrow \downarrow \uparrow \uparrow \uparrow \rangle$

Figure 4

Storage layout of the distributed Hilbert space. The prefixes are randomly distributed among the MPI processes using a hash function. States with same prefixes are mapped to the same process. Within a process, the states are ordered lexicographically. The Hamiltonian matrix is not stored.

Figure 5

Geometries of Heisenberg spin-$1/2$ model benchmarks. Different colors (symbols) show the sublattice structure used for the sublattice coding technique. The gray background shows the Wigner-Seitz cell defining the periodicity of the lattice. Left: Triangular lattice, 48 sites, four sublattices structure. Middle left: Square lattice, 48 sites, four sublattices structure. Middle right: kagome lattice, 48 sites, three sublattices structure. Right: Square lattice, 50 sites, five sublattices structure.

Figure 6

Scaling behavior of the parallelization. The time for one matrix-vector multiplication in seconds is compared for the total number of processes. Both axes are scaled logarithmically. The matrix is the Hamiltonian of the Heisenberg spin $1/2$ antiferromagnet on a 40-site cluster with additional next-nearest neighbor and third-nearest neighbor interactions for the benchmark on $16,64,128$, and 256 cores. For $1024,2048$, and 4096 cores the matrix is the Hamiltonian of the Heisenberg spin $1/2$ antiferromagnet on a 48-site triangular cluster. The Hamiltonian is considered in its ground state sector. We observe almost ideal scaling behavior up to 4096 processors.