Mixed Spin Chain Example¶

1. Introduction¶

This example illustrates how QS³-ED2 can treat systems with site-dependent spin magnitudes.

The local spin value at each site is defined in

FILE_SPIN = list_spin.dat

which allows different spins to be assigned to different lattice sites.

This example therefore demonstrates

mixed local Hilbert spaces
translation symmetry in enlarged unit cells
Lanczos ground-state calculation
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  = 50

Although the system contains 100 spins, the translation period is

\[ L_1 = 50 \]

because the unit cell contains two sites.

Thus the chain can be viewed as

\[ ( A_1,B_1,A_2,B_2,\dots,A_{50},B_{50} ) \]

with periodic boundary conditions.

Nearest-neighbor bonds

NO_TWO = 100

5. Symmetry Operations¶

Translational symmetry is defined by

FILE_S1 = list_p1.dat

The translation operator shifts the system by one unit cell

\[ (A_i,B_i) \rightarrow (A_{i+1},B_{i+1}). \]

6. Momentum (Wavevector) Sector¶

Wavevector parameters

L1 = 50
M1 = 0

Allowed momenta

\[ k = \frac{2\pi m}{L_1}, \qquad m=0,1,\dots,49. \]

The calculation selects

M1 = 0

which corresponds to

\[ k = 0. \]

Thus the ground state is computed in the zero-momentum sector

\[ T|\psi\rangle = |\psi\rangle. \]

7. Local Hilbert Space¶

The local spin values are defined in

list_spin.dat

This file specifies the spin magnitude at each site, enabling mixed-spin systems.

For example

S1 S2 S1 S2 ...

could represent an alternating

\[ S_1 = \frac12, \qquad S_2 = 1 \]

chain.

8. NOD Sector Restriction¶

QS³-ED2 uses

\[ n_i = S_i - m_i \]

and defines

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

Input parameters

NODmin = 0
NODmax = 3

restrict the allowed basis states.

The maximum allowed value depends on the spin magnitude at each site.

9. Hilbert-space Dimension¶

From the program output

THS   = 171801
THS(k)= 3438

THS : total basis states 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: 160

11. Ground-state Energy¶

The converged ground-state energy is

\[ E_0 = -5.652675156567411. \]

12. Eigenvector Accuracy¶

Verification

\[ \langle GS|H|GS\rangle = -5.652675156567412 \]

Residual

\[ |1-(\langle GS|H|GS\rangle)^2/\langle GS|H^2|GS\rangle| =5.551115123125783\times10^{-16}. \]

The solution therefore reaches 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

14. Runtime¶

Measured runtime

Time: 0.276 sec

15. Summary¶

This example demonstrates a mixed-spin quantum chain calculation with QS³-ED2.

The example highlights

heterogeneous local Hilbert spaces
symmetry reduction with enlarged unit cells
momentum-sector diagonalization
Lanczos ground-state computation
calculation of physical observables.