Capabilities · Ground state

Ground-state calculations

Worked examples for obtaining ground-state energies and eigenvectors with the Lanczos method in SCS_Lanczos.

Documentation Ground-state calculations

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Ground-state examples

The examples follow a common workflow—select the model and symmetry sector, prepare the input, run Lanczos, and inspect the converged ground state.

Detailed example

Spin model

Ground-state calculation using the Model_Spins implementation.

Open example → Detailed example

Hubbard model

One-orbital 1D Hubbard-chain ground state at half filling.

Open example →

Planned

t–J model

Projected-fermion ground state using the Model_1_orb_tJ implementation.

Example to be added

Worked example 02

1D Hubbard-chain ground state and T=0 observables

This is a complete, reproducible example copied from a tested run. The model is 1_orb_Hubbard_chain in LongRange geometry mode, with L=12, Nup=6, and Ndown=6.

Model1_orb_Hubbard_chain

GeometryLongRange

System sizeL = 12

SectorNup = 6, Ndown = 6

InteractionU = 16 on all sites

Hilbert-space size853776

Static spin-structure factor: varying U at L=12

At fixed L=12, the figure below compares U=0,2,4,6,8,10,16 at half filling (N=12) in the Sz=0 sector. The spin-structure factor S(q) uses SzSz + 0.5(SpSm+SmSp). The results were produced with the SCS_Lanczos with last commit id 060782140850e9b22ca43bcb0635cbefd3c025e5.

Static spin structure factor S(q) for the half-filled L=12 Hubbard chain at U=0,2,4,6,8,10,16 — Fixed `L=12`, varying `U=0,2,4,6,8,10,16`. View PDF.

Static spin-structure factor: varying L at U=10

At fixed U=10, the figure below compares chain lengths L=8,10,12,14,16 at half filling (N=L) in the Sz=0 sector.

Static spin structure factor S(q) for half-filled Hubbard chains at U=10 and L=8,10,12,14,16, with vertically offset curves — Fixed `U=10`, varying `L=8,10,12,14,16`. Curves are vertically offset for clarity; dashed lines mark their baselines. View PDF.

Example files

input_1orbHubbard_LongRange.inpMain exact input file.

Hopping_L12_chain_PBC.txtNearest-neighbor hopping matrix for PBC chain.

Local_U.txtOn-site interaction values (all sites set to 16).

ZeroMatrix_L12.txtZero nonlocal interactions/exchanges/pair-hopping matrix.

ThreePointOprs_zero.txtThree-point observation placeholder.

ThreePointIntrs_zero.txtThree-point interaction placeholder.

Sq_U16_fullSS.pdfPDF plot for publication-quality output.

README.mdDetailed notes and reproduction command.

To Run: ./lanczos input_1orbHubbard_LongRange.inp

Time for ground state and spin-spin correlations
L	N	Hilbert space dimension	Time (1 core)
12	12	853776	~33 sec
14	14	11778624	~11m 50sec
16	16	165636900	~3hr 15m

CPU architecture: The timing above was measured using one CPU core on PARAM Shakti at IIT Kharagpur, which uses Intel Xeon Gold 6148 (Skylake) processors at a base frequency of 2.4 GHz. Hardware specifications.

Worked example 01

Spin-model ground state

This example documents a complete spin-model calculation, from defining the Hamiltonian and lattice to checking the final ground-state energy. The exact input listing and commands will be filled using a tested SCS_Lanczos run.

1. Model route 2. Physical parameters 3. Input file 4. Run command 5. Expected output

Step 1

Define the spin model

Code implementationmodels/Model_Spins.*

HamiltonianTo be specified for this example

Lattice and sizeTo be specified

Boundary conditionTo be specified

Symmetry sectorTo be specified

The documentation will state all conventions explicitly, including site numbering, bond ordering, the sign and normalization of exchange couplings, and the selected spin sector.

Step 2

Prepare the input file

The tested input file will be reproduced here in full. Each active parameter will be explained directly below the listing.

# Spin-model input example
# A tested input file will be inserted here.

Parameters to document

Model selector

Number of sites

Spin or target S^z sector

Exchange connections and couplings

Lanczos convergence controls

Requested eigenvalues and eigenvectors

Step 3

Build and run

The exact compile target and executable invocation will be shown here after verifying the example against the current repository version.

# Build command
make

# Model-specific run command
# To be added from a tested run

Step 4

Read the output

The completed example will identify the lines reporting the Hilbert-space size, Lanczos iterations, convergence, ground-state energy, and saved eigenvector information.

Basis sizeConfirms the selected Hilbert space and symmetry sector.

IterationTracks convergence of the lowest Ritz value.

E0Ground-state energy for the chosen parameters.

Residual / toleranceConfirms numerical convergence.

Step 5

Validation checklist

Confirm that the basis dimension matches the intended spin sector.
Repeat with a tighter Lanczos tolerance and verify that the reported energy is stable.
For a small lattice, compare with the full exact-diagonalization result.
Check the energy against a known analytic limit or benchmark when available.
Record the code revision, input file, and output together for reproducibility.

Next documentation step

Complete the tested Spin-model example

To replace the placeholders, we will add one working input file, the corresponding run command, and a short representative output from SCS_Lanczos.