KPM

Capability

Kernel Polynomial Method (KPM) for various quantities:

KPM for density of states (DOS) (RevModPhys.78.275):

mu = KPM.kpm_1d(H_norm, NC, NR)
E, rho = KPM.dos(mu, D)

where H_norm is the scaled Hamiltonian with original half bandwidth D, NC is the expansion order and NR is the number of random vectors for stochastic estimation of trace in KPM. The function KPM.dos evaluate the DOS rho with energy points E.

KPM for DC conductivity (linear response) (RevModPhys.78.275):

μ2Dxy = KPM.kpm_2d(H_norm, Jx, Jy, NC, NR, NH)
dσxyE = KPM.d_dc_cond(μ2Dxy, D, Evals)

where Jx, Jy are current operators and NH is the dimensionality of the Hamiltonian H_norm. The function KPM.d_dc_cond evaluate the differential first-order conductivity dσxyE with given energy points Evals, integration of which (with Fermi distribution) gives DC conductivity at certain temperature.

KPM for frequency-dependent nonlinear response (arXiv:1810.03732):

mu_3d_xyz = KPM.kpm_3d(H_norm, Jx, Jy, Jz, NC, NR, NH)
dchi_xyz = KPM.d_cpge(mu_3d_xyz, NC, w1, w2, E)

where dchi_xyz is the differential second-order conductivity (arXiv:2312.14244), w1, w2 are two frequencies and E is the energy to evaluate, integration of which (with Fermi distribution) gives non at certain temperature.

Installation

First, install the old version of CUDA with

] add CUDA@3.12.0

Then, add the KPM package

] add https://github.com/Pixley-Research-Group-in-CMT/KPM.jl

If the code automatically upgrade CUDA, go to ~/.julia/packages/CUDA/ (or where you put your package) and delete the higher version of CUDA. Then, ] rm CUDA to uninstall CUDA from registries and ] add CUDA@3.12.0, which will also precompile KPM.

This is an unregistered package for now, so we need to use github URL to add package. Github username and password needed.

After installation, you should be able to import the package by

    julia> using KPM

without further action with github. To update, use

] update KPM

and type github username / password when prompted.

Getting started with DOS

You will need Hamiltonian (H_orig).

The most simple way to calculate DOS is

julia> E, rhoE = KPM.dos(H_orig)

For many situation, you may need more control over the calculation. In such case, you may calculate moment first and then convert to DOS. When calculating density of state, left and right input vectors for KPM need to be random and identical when both written as a ket. This is done by default in kpm_1d, or you may supply your own input vectors.

When using kpm_1d (and kpm_1d!, the in place version), Hamiltonian must be normalized. H_norm is the Hamiltonian normalized by a such that H_orig * a == H_norm, and H_norm has all eigenvalues in (-1, 1). Calculating mu and then DOS allows calculating DOS and its derivative at zero energy simpler.

julia> NC = 1024; NR = 13;
julia> mu = KPM.KPM_1d(H_norm, NC, NR)
julia> rho_0 = KPM.dos0(mu, a)
julia> d2rho_0 = KPM.dos0(mu, a; dE_order=2)
julia> E_grid, rho_E = dos(mu, a, 50; E_range=[-1.5, 1.5])

DC conductivity WIP:

Some practical tips:

• Store your github credentials (for convenience). See this instruction.
• When using Amarel, the interactive computing nodes do not have git available. Add/update package on login node.
• When using RUPC, ssh to one of n34, n35, n40. Add package there. Load it once to compile.
• If there is any permission issue when running, add permission: chmod -r +x ~/.julia/. For permission error when building, simply ignore.

Notes about GPU:

This package uses GPU automatically when GPU is available. No action needed. To check whether GPU is enabled, run

  KPM.whichcore()

It is also recommended to add this line right after importing KPM in julia code to guarantee using GPU correctly.