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J1–J2 Chain Example

Directory

examples/j1j2_chain/

This example demonstrates how to run QS³-ED2 for a frustrated one‑dimensional spin chain with both nearest‑neighbor (\(J_1\)) and next‑nearest‑neighbor (\(J_2\)) interactions.

The system consists of

\[ N = 100 \]

spin‑1/2 sites 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/j1j2_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.

1. Introduction

This example extends the basic chain calculation by introducing next‑nearest‑neighbor interactions, producing a frustrated spin system.

The example demonstrates

  • construction of a Hamiltonian with multiple bond ranges
  • symmetry reduction using lattice translations
  • momentum‑sector selection
  • Lanczos diagonalization
  • evaluation of physical observables.

2. Model Hamiltonian

The Hamiltonian is

\[ H = \sum_{\langle i,j\rangle_1} H^{(1)}_{ij} + \sum_{\langle i,j\rangle_2} H^{(2)}_{ij} + \sum_i \mathbf{h}\cdot\mathbf{S}_i \]

where the bond Hamiltonian is

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

The symmetric anisotropic interaction is

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

Here

  • \(n=1\) denotes nearest‑neighbor (\(J_1\)) interactions
  • \(n=2\) denotes next‑nearest‑neighbor (\(J_2\)) interactions.

3. Coupling Parameters

Magnetic field

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

Nearest‑neighbor (\(J_1\)) couplings

\[ (J_x^{(1)},J_y^{(1)},J_z^{(1)})=(1.0,0.8,0.7) \]
\[ \mathbf{D}^{(1)}=(0.2,0.1,5.0) \]
\[ (\Gamma_x^{(1)},\Gamma_y^{(1)},\Gamma_z^{(1)})=(0.1,0.3,-0.2) \]

Next‑nearest‑neighbor (\(J_2\)) couplings

\[ (J_x^{(2)},J_y^{(2)},J_z^{(2)})=(0.5,0.4,0.35) \]
\[ \mathbf{D}^{(2)}=(0.1,0.05,2.5) \]
\[ (\Gamma_x^{(2)},\Gamma_y^{(2)},\Gamma_z^{(2)})=(0.05,0.15,-0.1) \]

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

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

Next‑nearest‑neighbor bonds

\[ (1,3),(2,4),\dots,(100,2) \]

Total number of bonds

NO_TWO = 200

5. Symmetry Operations

Translational symmetry

FILE_S1 = list_p1.dat

Translation operator

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

This corresponds to the cyclic shift

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

6. Momentum (Wavevector) Sector

Wavevector parameters from the input

L1 = 100
M1 = 0

The allowed momenta are

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

The parameter

M1 = 0

selects

\[ k = 0. \]

Thus the calculation is performed in the zero‑momentum sector

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

7. Local Hilbert Space

Each 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 counter

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

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)= 1669
  • THS : dimension before symmetry reduction
  • THS(k) : dimension 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

\[ E_0 = -3.656548006870278 \times 10^{0}. \]

12. Eigenvector Accuracy

Verification

\[ \langle GS|H|GS\rangle = -3.656548006870279 \]

Residual

\[ |1-(\langle GS|H|GS\rangle)^2/\langle GS|H^2|GS\rangle| =7.771561172376096\times10^{-16}. \]

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 correlation tensor
two_body_cf_z+-.dat ladder correlations

Correlation pairs are defined in

list_ij_cf.dat

Example

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

14. Runtime

Measured runtime

Time: 0.329 sec

15. Summary

This example demonstrates a frustrated \(J_1\)–\(J_2\) spin chain calculation using QS³‑ED2.

Key features illustrated include

  • multi‑range spin interactions
  • translational symmetry reduction
  • momentum‑sector selection
  • Lanczos ground‑state computation
  • evaluation of correlation functions.