1D Chain Example

Directory

examples/chain/

This example demonstrates how to run QS³‑ED2 for a periodic one‑dimensional spin system.

The system consists of

\[ N = 100 \]

spin‑1/2 sites arranged on a periodic chain.

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/chain/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.

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1. Introduction

This example illustrates a basic calculation with QS³‑ED2 for a periodic one‑dimensional quantum spin system.

The workflow demonstrates

• construction of the Hamiltonian
• symmetry reduction using lattice translations
• momentum‑sector selection
• Lanczos diagonalization
• evaluation of physical observables.

The model includes

• anisotropic exchange (XYZ)
• Dzyaloshinskii–Moriya interaction
• symmetric anisotropic \(\Gamma\) interaction
• uniform magnetic field.

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2. Model Hamiltonian

The Hamiltonian is

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

where

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

The symmetric anisotropic interaction is

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

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3. Coupling Parameters

In this example the couplings are uniform.

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

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

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4. Lattice Structure

System parameters from output.dat

NOS = 100
L1  = 100
L2 = L3 = L4 = L5 = L6 = 1

Thus the system is a periodic one‑dimensional chain.

Nearest‑neighbor bonds

\[ (1,2),(2,3),\dots,(100,1). \]

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5. Symmetry Operations

Translational symmetry is defined by

FILE_S1 = list_p1.dat

The translation operator is

\[ T(i)=i+1 \quad (i=1,\dots,99), \qquad T(100)=1. \]

This corresponds to the cyclic shift

\[ (S_1,S_2,\dots,S_{100}) \rightarrow (S_2,S_3,\dots,S_{100},S_1). \]

This translation generates the cyclic symmetry group of the periodic chain.

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6. Momentum (Wavevector) Sector

QS³‑ED2 allows diagonalization within a fixed crystal momentum sector defined with respect to the translation symmetry.

The input parameters

M1 = 0

select the momentum sector corresponding to the eigenvalue

\[ T |\psi\rangle = e^{i k} |\psi\rangle . \]

For a chain of length

\[ L_1 = 100 \]

the allowed wavevectors are

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

The parameter

M1 = 0

therefore selects

\[ k = 0. \]

Thus the calculation is performed in the zero‑momentum sector, i.e.

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

This sector contains the translationally invariant states and typically hosts the ground state for many spin models.

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7. Local Hilbert Space

Each site hosts a spin

\[ S=\frac12 \]

so the local Hilbert‑space dimension is

\[ 2S+1=2. \]

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8. NOD Sector Restriction

QS³‑ED2 uses the integer

\[ n_i = S_i - m_i \]

For spin‑1/2

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

The global counter

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

equals the number of down spins.

Input parameters

NODmin = 0
NODmax = 3

restrict

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

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9. Hilbert‑space Dimension

From the program output

THS   = 166751
THS(k)= 1669

• THS : dimension before symmetry reduction
• THS(k) : representative states after symmetry and momentum reduction

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10. Lanczos Solver

Solver parameters

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

Total Lanczos iterations

total lanczos step: 150

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11. Ground‑state Energy

The converged ground‑state energy is

\[ E_0 = -1.299620173300453 \times 10^{1}. \]

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12. Eigenvector Accuracy

Verification printed by the program

\[ \langle GS|H|GS\rangle = -1.299620173300453 \times 10^{1} \]

Residual

\[ |1-(\langle GS|H|GS\rangle)^2 / \langle GS|H^2|GS\rangle| =4.440892098500626 \times 10^{-16} \]

This indicates convergence close to machine precision.

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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	full spin correlations
two_body_cf_z+-.dat	ladder‑operator correlations

Correlation pairs are defined in

list_ij_cf.dat

Example

\[ (1,2),(1,3),\dots,(1,10). \]

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14. Runtime

Measured runtime

Time: 0.221 sec

The runtime depends mainly on

• Hilbert‑space dimension
• number of Lanczos iterations
• BLAS performance.

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15. Summary

This example demonstrates a QS³‑ED2 calculation for a periodic spin chain.

Key features illustrated here include

• construction of anisotropic spin Hamiltonians
• translational symmetry reduction
• momentum‑sector selection
• Lanczos ground‑state computation
• evaluation of magnetization and correlation functions.