Triangular Lattice Example¶

1. Introduction¶

This example illustrates a frustrated two-dimensional lattice model using QS³-ED2.

The triangular lattice introduces geometric frustration because each site interacts with six nearest neighbors.

The example demonstrates

Hamiltonian construction on a frustrated lattice
translational symmetry in two spatial directions
momentum-sector selection
Lanczos diagonalization
evaluation of observables.

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 = 100
L1 = 10
L2 = 10

The system forms a

\[ 10 \times 10 \]

triangular lattice with periodic boundary conditions.

Each site has six nearest neighbors.

Total number of bonds

NO_TWO = 300

5. Symmetry Operations¶

Translational symmetry is defined by

FILE_S1 = list_p1.dat
FILE_S2 = list_p2.dat

These correspond to lattice translations

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

\[ T_y : (x,y) \rightarrow (x,y+1). \]

Periodic boundary conditions are applied in both directions.

6. Momentum (Wavevector) Sector¶

Wavevector parameters

L1 = 10
L2 = 10
M1 = 0
M2 = 0

Allowed wavevectors are

\[ k_x = \frac{2\pi m_1}{L_1}, \qquad k_y = \frac{2\pi m_2}{L_2} \]

with

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

The input selects

M1 = 0
M2 = 0

which corresponds to

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

Thus the calculation is performed in the zero-momentum sector

\[ T_x|\psi\rangle = |\psi\rangle, \qquad T_y|\psi\rangle = |\psi\rangle. \]

7. Local Hilbert Space¶

Each lattice site hosts

\[ S=\frac12 \]

so the local Hilbert-space dimension is

\[ 2S+1=2. \]

8. NOD Sector Restriction¶

QS³-ED2 labels basis states using

\[ n_i = S_i - m_i \]

For spin-1/2

\[ n_i = \begin{cases} 0 & (m_i=+1/2) \\ 1 & (m_i=-1/2) \end{cases} \]

The global quantity

\[ \mathrm{NOD}=\sum_i n_i \]

counts the number of down spins.

Input parameters

NODmin = 0
NODmax = 3

restrict

\[ N_\downarrow \in \{0,1,2,3\}. \]

9. Hilbert-space Dimension¶

From output.dat

THS   = 166751
THS(k)= 1670

THS : Hilbert-space dimension before symmetry reduction
THS(k) : representative states after symmetry and momentum reduction

10. Lanczos Solver¶

Solver parameters

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

Total Lanczos iterations

total lanczos step: 110

11. Ground-state Energy¶

The converged ground-state energy is

\[ E_0 = 7.186891958806912. \]

12. Eigenvector Accuracy¶

Verification

\[ \langle GS|H|GS\rangle = 7.186891958806907 \]

Residual

\[ |1-(\langle GS|H|GS\rangle)^2/\langle GS|H^2|GS\rangle| =9.992007221626409\times10^{-16}. \]

This indicates convergence close to machine precision.

13. 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

Correlation pairs are specified in

list_ij_cf.dat

14. Runtime¶

Measured runtime

Time: 0.318 sec

15. Summary¶

This example demonstrates a triangular-lattice quantum spin model calculation using QS³-ED2.

Key features illustrated include

Hamiltonians on frustrated lattices
two-dimensional translational symmetry
momentum-sector diagonalization
Lanczos ground-state computation
calculation of magnetization and correlation functions.