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Cubic Lattice (Sparse Hilbert Space, Fixed NOD) Example

Directory

examples/cubic_sp_HB/

This example demonstrates how to run QS³‑ED2 for a three‑dimensional cubic lattice with periodic boundary conditions in a fixed NOD sector and using additional reflection symmetries.

The system contains

\[ N = 1000 \]

spin‑1/2 sites arranged on a

\[ 10 \times 10 \times 10 \]

cubic lattice.

The ground state is computed using the Lanczos method, and the program evaluates

  • ground‑state energy
  • local magnetization
  • two‑point spin correlations.

Note

The numerical values shown in this document are taken from the reference output stored in

examples/ref_dat/cubic_sp_HB/output.dat.

These files are provided as reference data for documentation and regression testing. The exact numerical values may vary slightly depending on the compilation environment and hardware.

1. Introduction

This example illustrates a large‑scale 3D lattice calculation using QS³‑ED2, where the computation is restricted to a single NOD sector and further reduced using translational and reflection symmetries.

Key features include

  • cubic lattice geometry in three dimensions
  • translational symmetry in three directions
  • reflection symmetry in three directions
  • momentum‑sector selection
  • fixed‑\(\mathrm{NOD}\) restriction
  • Lanczos diagonalization in a reduced Hilbert space.

2. Model Hamiltonian

The Hamiltonian is

\[ H = \sum_{\langle i,j\rangle} H_{ij} + \sum_i \mathbf{h}\cdot\mathbf{S}_i \]

with bond interaction

\[ H_{ij} = \sum_{\alpha=x,y,z} J_\alpha S_i^\alpha S_j^\alpha + \mathbf{D}\cdot(\mathbf{S}_i \times \mathbf{S}_j) + H_\Gamma(i,j). \]

The symmetric anisotropic interaction is

\[ H_\Gamma(i,j)= \Gamma_x(S_i^y S_j^z + S_i^z S_j^y) + \Gamma_y(S_i^z S_j^x + S_i^x S_j^z) + \Gamma_z(S_i^x S_j^y + S_i^y S_j^x). \]

3. Coupling Parameters

Magnetic field

\[ \mathbf{h}=(-0.1,-0.1,-0.3) \]

Exchange parameters

\[ (J_x,J_y,J_z)=(1.0,0.8,0.7) \]

Dzyaloshinskii–Moriya interaction

\[ \mathbf{D}=(0.2,0.1,5.0) \]

\(\Gamma\) interaction

\[ (\Gamma_x,\Gamma_y,\Gamma_z)=(0.1,0.3,-0.2) \]

4. Lattice Structure

System parameters from output.dat

NOS = 1000
L1 = 10
L2 = 10
L3 = 10

For a cubic lattice

\[ N = L_1 L_2 L_3 = 10\times10\times10 = 1000 \]

Each site has coordination number

\[ z = 6 \]

Total number of nearest‑neighbor bonds

NO_TWO = 3000

which matches

\[ \frac{Nz}{2} = \frac{1000\times6}{2} = 3000. \]

5. Translational Symmetry

Translational symmetry operations are defined by

FILE_S1 = list_p1.dat
FILE_S2 = list_p2.dat
FILE_S3 = list_p3.dat

corresponding to

\[ T_x : (x,y,z) \rightarrow (x+1,y,z) \]
\[ T_y : (x,y,z) \rightarrow (x,y+1,z) \]
\[ T_z : (x,y,z) \rightarrow (x,y,z+1) \]

with periodic boundary conditions.

6. Momentum (Wavevector) Sector

Wavevector parameters

L1 = 10
L2 = 10
L3 = 10
M1 = 0
M2 = 0
M3 = 0

Allowed wavevectors

\[ k_x = \frac{2\pi m_1}{L_1},\quad k_y = \frac{2\pi m_2}{L_2},\quad k_z = \frac{2\pi m_3}{L_3} \]

with

\[ m_1,m_2,m_3 = 0,\dots,9. \]

The present calculation selects

\[ (k_x,k_y,k_z)=(0,0,0). \]

7. Reflection (Inversion) Symmetry

In addition to translations, QS³‑ED2 can use reflection operations along each lattice direction.

The parameters

L4 = 2
L5 = 2
L6 = 2

activate reflection operations for the

  • x direction
  • y direction
  • z direction

respectively.

These operations correspond to

\[ R_x : (x,y,z) \rightarrow (-x,y,z) \]
\[ R_y : (x,y,z) \rightarrow (x,-y,z) \]
\[ R_z : (x,y,z) \rightarrow (x,y,-z). \]

The corresponding quantum numbers are specified by

M4 = 0
M5 = 0
M6 = 0

which select the even parity sector under each reflection.

Thus the ground state satisfies

\[ R_x|\psi\rangle = |\psi\rangle \]
\[ R_y|\psi\rangle = |\psi\rangle \]
\[ R_z|\psi\rangle = |\psi\rangle. \]

Using these reflection symmetries significantly reduces the Hilbert‑space size.

8. Local Hilbert Space

Each site hosts

\[ S=\frac12 \]

so the local Hilbert‑space dimension is

\[ 2S+1=2. \]

9. NOD Sector Restriction

The NOD restriction is fixed

NODmin = 3
NODmax = 3

Thus

\[ \mathrm{NOD} = 3. \]

For spin‑1/2 systems this corresponds to

\[ N_\downarrow = 3. \]

10. Hilbert‑space Dimension

From output.dat

THS   = 166167000
THS(k)= 23719
  • THS : dimension before symmetry reduction
  • THS(k) : representative states after symmetry reduction

11. Lanczos Solver

The Lanczos work‑space parameter is fixed

MNTE = 19

Output confirms

Current MNTE = 19
Optimal MNTE = 19

Solver parameters

LNC_ENE_CONV = 1.0E-14
MAXITR = 10000
MINITR = 5
ITRINT = 5

Total iterations

total lanczos step: 140

12. Ground‑state Energy

The converged ground‑state energy is

\[ E_0 = 7.367224213680655 \times 10^{2}. \]

13. Eigenvector Accuracy

Verification

\[ \langle GS|H|GS\rangle = 7.367224213680651 \times 10^{2} \]

Residual

\[ |1-(\langle GS|H|GS\rangle)^2/\langle GS|H^2|GS\rangle| =1.110223024625157\times10^{-15}. \]

14. Observables

Enabled in the input

CAL_LM = 1
CAL_CF = 1

Generated files

file description
local_mag.dat local magnetization
two_body_cf_xyz.dat spin correlations
two_body_cf_z+-.dat ladder correlations

15. Runtime

Measured runtime

Time: 121.074 sec

16. Summary

This example demonstrates a large‑scale

\[ 10\times10\times10 \]

cubic‑lattice calculation with QS³‑ED2 using

  • translational symmetry
  • reflection symmetry
  • fixed‑\(\mathrm{NOD}\) restriction
  • Lanczos diagonalization
  • symmetry‑reduced Hilbert spaces.