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pyqula

Python library for quantum lattice tight binding models

0.0.76
PyPI

Maintainers: 1

SUMMARY

This is a Python library to compute quantum-lattice tight-binding models in different dimensionalities.

INSTALLATION

With pip (release version)

pip install --upgrade pyqula

Manual installation (most recent version)

Clone the Github repository with

git clone https://github.com/joselado/pyqula

and add the "pyqula/src" path to your Python script with

import sys
sys.path.append(PATH_TO_PYQULA+"/src")

Tutorials

Jupyter notebooks with tutorials can be found in the links below (part of the Jyvaskyla Summer School 2022 )

FUNCTIONALITIES

Single particle Hamiltonians

Spinless, spinful and Nambu basis for orbitals
Full non-collinear electron and Nambu formalism
Include magnetism, spin-orbit coupling and superconductivity
Band structures with state-resolved expectation values
Momentum-resolved spectral functions
Local and full operator-resolved density of states
0d, 1d, 2d and 3d tight binding models
Electronic structure unfolding in supercells

Interacting mean-field Hamiltonians

Selfconsistent mean-field calculations with local/non-local interactions
Both collinear and non-collinear formalism
Anomalous mean-field for non-collinear superconductors
Full selfconsistency with all Wick terms for non-collinear superconductors
Constrained and unconstrained mean-field calculations
Automatic identification of order parameters for symmetry broken states
Hermitian and non-Hermitian mean-field calculations

Topological characterization

Berry phases, Berry curvatures, Chern numbers and Z2 invariants
Operator-resolved Chern numbers and Berry density
Frequency resolved topological density
Spatially resolved topological flux
Real-space Chern density for amorphous systems
Wilson loop and Green's function formalism

Spectral functions

Spectral functions in infinite geometries
Surface spectral functions for semi-infinite systems
Interfacial spectral function in semi-infinite junctions
Single impurities in infinite systems
Operator-resolved spectral functions
Green's function renormalization algorithm

Chebyshev kernel polynomial based-algorithms

Local and full spectral functions
Non-local correlators and Green's functions
Locally resolved expectation values
Operator resolved spectral functions
Reaching system sizes up to 10000000 atoms on a single-core laptop

Quantum transport

Metal-metal transport
Metal-superconductor transport
Fully non-collinear Nambu basis
Non-equilibrium Green's function formalism
Operator-resolved transport
Differential decay rate
Tunneling and contact scanning probe spectroscopy

EXAMPLES

A variety of examples can be found in pyqula/examples. Short examples are shown below

Band structure of a Kagome lattice

from pyqula import geometry
g = geometry.kagome_lattice() # get the geometry object
h = g.get_hamiltonian() # get the Hamiltonian object
(k,e) = h.get_bands() # compute the band structure

Alt text

Valley-resolved band structure of a honeycomb superlattice

from pyqula import geometry
g = geometry.honeycomb_lattice() # get the geometry object
g = g.get_supercell(7) # create a supercell
h = g.get_hamiltonian() # get the Hamiltonian object
(k,e,v) = h.get_bands(operator="valley") # compute the band structure

Alt text

Interaction-driven spin-singlet superconductivity

from pyqula import geometry
import numpy as np
g = geometry.triangular_lattice() # geometry of a triangular lattice
h = g.get_hamiltonian()  # get the Hamiltonian
h.setup_nambu_spinor() # setup the Nambu form of the Hamiltonian
h = h.get_mean_field_hamiltonian(U=-1.0,filling=0.15,mf="swave") # perform SCF
# electron spectral-function
h.get_kdos_bands(operator="electron",nk=400,energies=np.linspace(-1.0,1.0,100))

Alt text

Interaction driven non-unitary spin-triplet superconductor

import numpy as np
from pyqula import geometry
g = geometry.triangular_lattice() # generate the geometry
h = g.get_hamiltonian() # create Hamiltonian of the system
h.add_exchange([0.,0.,1.]) # add exchange field
h.setup_nambu_spinor() # initialize the Nambu basis
# perform a superconducting non-collinear mean-field calculation
h = h.get_mean_field_hamiltonian(V1=-1.0,filling=0.3,mf="random")
# compute the non-unitarity of the spin-triplet superconducting d-vector
d = h.get_dvector_non_unitarity() # non-unitarity of spin-triplet
# electron spectral-function
h.get_kdos_bands(operator="electron",nk=400,energies=np.linspace(-2.0,2.0,400))

Alt text

Mean-field with local interactions of a zigzag honeycomb ribbon

from pyqula import geometry
g = geometry.honeycomb_zigzag_ribbon(10) # create geometry of a zigzag ribbon
h = g.get_hamiltonian() # create hamiltonian of the system
h = h.get_mean_field_hamiltonian(U=1.0,filling=0.5,mf="ferro")
(k,e,sz) = h.get_bands(operator="sz") # calculate band structure

Alt text

Non-collinear mean-field with local interactions of a square lattice

from pyqula import geometry
g = geometry.square_lattice() # geometry of a square lattice
g = g.get_supercell([2,2]) # generate a 2x2 supercell
h = g.get_hamiltonian() # create hamiltonian of the system
h.add_zeeman([0.,0.,0.1]) # add out-of-plane Zeeman field
h = h.get_mean_field_hamiltonian(U=2.0,filling=0.5,mf="random") # perform SCF
(k,e,c) = h.get_bands(operator="sz") # calculate band structure
m = h.get_magnetization() # get the magnetization

Alt text

Interaction-induced non-collinear magnetism in a defective square lattice with spin-orbit coupling

from pyqula import geometry
g = geometry.square_lattice() # geometry of a square lattice
g = g.get_supercell([7,7]) # generate a 7x7 supercell
g = g.remove(i=g.get_central()[0]) # remove the central site
h = g.get_hamiltonian() # create hamiltonian of the system
h.add_rashba(.4) # add Rashba spin-orbit coupling
h = h.get_mean_field_hamiltonian(U=2.0,filling=0.5,mf="random") # perform SCF
(k,e,c) = h.get_bands(operator="sz") # calculate band structure
m = h.get_magnetization() # get the magnetization

Alt text

Band structure of twisted bilayer graphene

from pyqula import specialhamiltonian # special Hamiltonians library
h = specialhamiltonian.twisted_bilayer_graphene() # TBG Hamiltonian
(k,e) = h.get_bands() # compute band structure

Alt text

Band structure of monolayer NbSe2

from pyqula import specialhamiltonian # special Hamiltonians library
h = specialhamiltonian.NbSe2(soc=0.5) # NbSe2 Hamiltonian
(k,e,c) = h.get_bands(operator="sz",kpath=["G","K","M","G"]) # compute bands

Alt text

Chern number of an artificial Chern insulator

from pyqula import geometry
g = geometry.honeycomb_lattice()
h = g.get_hamiltonian()
h.add_rashba(0.2) # Rashba spin-orbit coupling
h.add_zeeman([0.,0.,0.6]) # Zeeman field
from pyqula import topology
(kx,ky,omega) = h.get_berry_curvature() # compute Berry curvature
c = h.get_chern() # compute the Chern number

Alt text

Topological phase transition in an artificial topological superconductor

import numpy as np
from pyqula import geometry
g = geometry.chain() # create a chain
g = g.supercell(100) # create a large supercell
g.dimensionality = 0 # make it finite
for J in np.linspace(0.,0.2,50): # loop over exchange couplings
    h = g.get_hamiltonian() # create a new hamiltonian
    h.add_onsite(2.0) # shift the chemical potential
    h.add_rashba(.3) # add rashba spin-orbit coupling
    h.add_exchange([0.,0.,J]) # add exchange coupling
    h.add_swave(.1) # add s-wave superconductivity
    edge = h.get_operator("location",r=g.r[0]) # projector on the edge
    energies = np.linspace(-.2,.2,200) # set of energies
    (e0,d0) = h.get_dos(operator=edge,energies=energies,delta=2e-3) # edge DOS

Alt text

Spatial distribution of Majorana modes in an artificial topological superconductor

import numpy as np
from pyqula import geometry
g = geometry.chain() # create a chain
g = g.supercell(100) # create a large supercell
g.dimensionality = 0 # make it finite
h = g.get_hamiltonian() # create a new hamiltonian
h.add_onsite(2.0) # shift the chemical potential
h.add_rashba(.3) # add rashba spin-orbit coupling
h.add_exchange([0.,0.,0.15]) # add exchange coupling
h.add_swave(.1) # add s-wave superconductivity
energies = np.linspace(-.15,.15,200) # set of energies
for ri in g.r: # loop over sites
    edge = h.get_operator("location",r=ri) # projector on that site
    (e0,d0) = h.get_dos(operator=edge,energies=energies,delta=2e-3) # local DOS

Alt text

Unfolded electronic structure of a supercell with a defect

from pyqula import geometry
import numpy as np
g = geometry.honeycomb_lattice() # create a honeycomb lattice
n = 3 # size of the supercell
g = g.get_supercell(n,store_primal=True) # create a supercell
h = g.get_hamiltonian() # get the Hamiltonian
fons = lambda r: (np.sum((r - g.r[0])**2)<1e-2)*100 # onsite in the impurity
h.add_onsite(fons) # add onsite energy
kpath = np.array(g.get_kpath(nk=200))*n # enlarged k-path
h.get_kdos_bands(operator="unfold",delta=1e-1,kpath=kpath) # unfolded bands

Alt text

Moire band structure of a moire superlattice

from pyqula import geometry
from pyqula import potentials
g = geometry.triangular_lattice() # create geometry
g = g.get_supercell([7,7]) # create a supercell
h = g.get_hamiltonian() # get the Hamiltonian
fmoire = potentials.commensurate_potential(g,n=3,minmax=[0,1]) # moire potential
h.add_onsite(fmoire) # add onsite energy following the moire
h.get_bands(operator=fmoire) # project on the moire

Alt text

Unfolded electronic structure of a moire superstructure

from pyqula import geometry
from pyqula import potentials
import numpy as np
g0 = geometry.triangular_lattice() # create geometry
n = 5 # supercell
g = g0.get_supercell(n,store_primal=True) # create a supercell
h = g.get_hamiltonian() # get the Hamiltonian
fmoire = potentials.commensurate_potential(g,n=3,minmax=[0,1]) # moire potential
h.add_onsite(fmoire) # add onsite energy following the moire
kpath = np.array(g.get_kpath(nk=400))*n # enlarged k-path
h.get_kdos_bands(operator="unfold",delta=2e-2,kpath=kpath,
                  energies=np.linspace(-3,-1,300)) # unfolded bands

Alt text

Band structure of a nodal line semimetal slab

from pyqula import geometry
from pyqula import films
g = geometry.diamond_lattice()
g = films.geometry_film(g,nz=20)
h = g.get_hamiltonian()
(k,e) = h.get_bands()

Alt text

Band structure of a three dimensional topological insulator slab

from pyqula import geometry
from pyqula import films
import numpy as np
g = geometry.diamond_lattice() # create a diamond lattice
g = films.geometry_film(g,nz=60) # create a thin film
h = g.get_hamiltonian() # generate Hamiltonian
h.add_strain(lambda r: 1.+abs(r[2])*0.8,mode="directional") # add axial strain
h.add_kane_mele(0.1) # add intrinsic spin-orbit coupling
(k,e,c)= h.get_bands(operator="surface") # compute band structure

Alt text

Surface spectral function of a 2D quantum spin-Hall insulator

from pyqula import geometry
g = geometry.honeycomb_lattice() # create a honeycomb lattice
h = g.get_hamiltonian() # generate Hamiltonian
h.add_soc(0.15) # add intrinsic spin-orbit coupling
h.add_rashba(0.1) # add Rashba spin-orbit coupling
h.get_surface_kdos(delta=1e-2) # compute surface spectral function

Alt text

Local density of states with atomic orbitals of a honeycomb nanoisland

from pyqula import islands
g = islands.get_geometry(name="honeycomb",n=3,nedges=3) # get an island
h = g.get_hamiltonian() # get the Hamiltonian
h.get_multildos(projection="atomic") # get the LDOS

Alt text

Interaction-driven magnetism in a honeycomb nanoisland

from pyqula import islands
g = islands.get_geometry(name="honeycomb",n=3,nedges=3) # get an island
h = g.get_hamiltonian() # get the Hamiltonian
h = h.get_mean_field_hamiltonian(U=1.0,filling=0.5,mf="ferro") # perform SCF
m = h.get_magnetization() # get the magnetization in each site

Alt text

Hofstadter's butterfly of a square lattice

import numpy as np
from pyqula import geometry
g = geometry.square_ribbon(40) # create square ribbon geometry

for B in np.linspace(0.,1.0,300): # loop over magnetic field
    h = g.get_hamiltonian() # create a new hamiltonian
    h.add_orbital_magnetic_field(B) # add an orbital magnetic field
    # calculate DOS projected on the bulk
    (e,d) = h.get_dos(operator="bulk",energies=np.linspace(-4.5,4.5,200))

Alt text

Landau levels of a Dirac semimetal

import numpy as np
from pyqula import geometry
g = geometry.honeycomb_ribbon(30) # create a honeycomb ribbon

for B in np.linspace(0.,0.02,100): # loop over magnetic field
    h = g.get_hamiltonian() # create a new hamiltonian
    h.add_orbital_magnetic_field(B) # add an orbital magnetic field
    # calculate DOS projected on the bulk
    (e,d) = h.get_dos(operator="bulk",energies=np.linspace(-1.0,1.0,200),
                       delta=1e-2)

Alt text

Surface spectral function of a Chern insulator

from pyqula import geometry
from pyqula import kdos
g = geometry.honeycomb_lattice() # create honeycomb lattice
h = g.get_hamiltonian() # create hamiltonian of the system
h.add_haldane(0.05) # Add Haldane coupling
kdos.surface(h) # surface spectral function

Alt text

Surface states and Berry curvature of a artificial 2D topological superconductor

from pyqula import geometry
import numpy as np
g = geometry.triangular_lattice() # get the geometry
h = g.get_hamiltonian() # get the Hamiltonian
h.add_onsite(2.0) # shift chemical potential
h.add_rashba(1.0) # Rashba spin-orbit coupling
h.add_zeeman([0.,0.,0.6]) # Zeeman field
h.add_swave(.3) # add superconductivity
(kx,ky,omega) = h.get_berry_curvature() # compute Berry curvature
h.get_surface_kdos(energies=np.linspace(-.4,.4,300)) # surface spectral function

Alt text

Surface states in a topological superconductor nanoisland

from pyqula import islands
g = islands.get_geometry(name="triangular",shape="flower",
                           r=14.2,dr=2.0,nedges=6) # get a flower-shaped island
h = g.get_hamiltonian() # get the Hamiltonian
h.add_onsite(3.0) # shift chemical potential
h.add_rashba(1.0) # Rashba spin-orbit coupling
h.add_zeeman([0.,0.,0.6]) # Zeeman field
h.add_swave(.3) # add superconductivity
h.get_ldos() # Spatially resolved DOS

Alt text

Antiferromagnet-superconductor interface

from pyqula import geometry
g = geometry.honeycomb_zigzag_ribbon(20) # create geometry of a zigzag ribbon
h = g.get_hamiltonian(has_spin=True) # create hamiltonian of the system
h.add_antiferromagnetism(lambda r: (r[1]>0)*0.5) # add antiferromagnetism
h.add_onsite(lambda r: (r[1]>0)*0.3) # add chemical potential
h.add_swave(lambda r: (r[1]<0)*0.3) # add superconductivity
(k,e,sz) = h.get_bands(operator="sz") # calculate band structure

Alt text

Fermi surface of a triangular lattice supercell

from pyqula import geometry
import numpy as np
g = geometry.triangular_lattice() # create geometry of the system
g = g.get_supercell(2) # create a supercell
h = g.get_hamiltonian() # create hamiltonian of the system
h.get_multi_fermi_surface(energies=np.linspace(-4,4,100),delta=1e-1)

Alt text

Unfolded Fermi surface of a supercell with a defect

from pyqula import geometry
import numpy as np
g0 = geometry.triangular_lattice()
n = 3 # size of the supercell
g = g0.get_supercell(n,store_primal=True) # create a supercell
h = g.get_hamiltonian() # get the Hamiltonian
fons = lambda r: (np.sum((r - g.r[0])**2)<1e-2)*100 # onsite in the impurity
h.add_onsite(fons) # add onsite energy
kpath = np.array(g.get_kpath(nk=200))*n # enlarged k-path
h.get_multi_fermi_surface(nk=50,energies=np.linspace(-4,4,100),
        delta=0.1,nsuper=n,operator="unfold")

Alt text

Tunneling and Andreev reflection in a metal-superconductor junction

from pyqula import geometry
from pyqula import heterostructures
import numpy as np
g = geometry.chain() # create the geometry
h = g.get_hamiltonian() # create the Hamiltonian
h1 = h.copy() # first lead
h2 = h.copy() # second lead
h2.add_swave(.01) # the second lead is superconducting
es = np.linspace(-.03,.03,100) # set of energies for dIdV
for T in np.linspace(1e-3,1.0,6): # loop over transparencies
    HT = heterostructures.build(h1,h2) # create the junction
    HT.set_coupling(T) # set the coupling between the leads
    Gs = [HT.didv(energy=e) for e in es] # calculate conductance

Alt text

From tunneling to contact transport of a topological superconductor

from pyqula import geometry
from pyqula import heterostructures
import numpy as np

g = geometry.chain() # create the geometry
h = g.get_hamiltonian() # create teh Hamiltonian
h1 = h.copy() # first lead
h2 = h.copy() # second lead
h2.add_onsite(2.0) # shift chemical potential in the second lead
h2.add_exchange([0.,0.,.3]) # add exchange in the second lead
h2.add_rashba(.3) # add Rashba SOC in the second lead
h2.add_swave(.05) # add s-wave SC in the second lead
es = np.linspace(-.1,.1,100) # grid of energies
for T in np.linspace(1e-3,0.5,10): # loop over transparencies
    HT = heterostructures.build(h1,h2) # create the junction
    HT.set_coupling(T) # set the coupling between the leads
    Gs = [HT.didv(energy=e) for e in es] # calculate transmission

Alt text

Single impurity in an infinite honeycomb lattice

from pyqula import geometry
import numpy as np
from pyqula import embedding
g = geometry.honeycomb_lattice() # create geometry 
h = g.get_hamiltonian() # get the Hamiltonian
hv = h.copy() # copy Hamiltonian to create a defective one
hv.add_onsite(lambda r: (np.sum((r - g.r[0])**2)<1e-2)*100) # add a defect
eb = embedding.Embedding(h,m=hv) # create an embedding object
(x,y,d) = eb.ldos(nsuper=19,energy=0.,delta=1e-2) # compute LDOS

Alt text

Bound state of a single magnetic impurity in an infinite superconductor

from pyqula import geometry
import numpy as np
from pyqula import embedding
g = geometry.square_lattice() # create geometry
h = g.get_hamiltonian() # get the Hamiltonian
h.add_swave(0.1) # add s-wave superconductivity
h.add_onsite(3.0) # shift chemical potential
hv = h.copy() # copy Hamiltonian to create a defective one
hv.add_exchange(lambda r: [0.,0.,(np.sum((r - g.r[0])**2)<1e-2)*6.]) # add magnetic site
eb = embedding.Embedding(h,m=hv) # create an embedding object
ei = eb.get_energy_ingap_state() # get energy of the impurity state
(x,y,d) = eb.ldos(nsuper=19,energy=ei,delta=1e-3) # compute LDOS

Alt text

Parity switching of a magnetic impurity in an infinite superconductor

from pyqula import geometry
from pyqula import embedding
import numpy as np

g = geometry.square_lattice() # create geometry
for J in np.linspace(0.,4.0,100): # loop over exchange
    h = g.get_hamiltonian() # get the Hamiltonian,spinless
    h.add_onsite(3.0) # shift chemical potential
    h.add_swave(0.2) # add s-wave superconductivity
    hv = h.copy() # copy Hamiltonian to create a defective one
    # add magnetic site
    hv.add_exchange(lambda r: [0.,0.,(np.sum((r - g.r[0])**2)<1e-2)*J])
    eb = embedding.Embedding(h,m=hv) # create an embedding object
    energies = np.linspace(-0.4,0.4,100) # energies
    d = [eb.dos(nsuper=2,delta=1e-2,energy=ei) for ei in energies] # compute DOS

Alt text

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Install

pyqula

SUMMARY

INSTALLATION

With pip (release version)

Manual installation (most recent version)

Tutorials

FUNCTIONALITIES

Single particle Hamiltonians

Interacting mean-field Hamiltonians

Topological characterization

Spectral functions

Chebyshev kernel polynomial based-algorithms

Quantum transport

EXAMPLES

Band structure of a Kagome lattice

Valley-resolved band structure of a honeycomb superlattice

Interaction-driven spin-singlet superconductivity

Interaction driven non-unitary spin-triplet superconductor

Mean-field with local interactions of a zigzag honeycomb ribbon

Non-collinear mean-field with local interactions of a square lattice

Interaction-induced non-collinear magnetism in a defective square lattice with spin-orbit coupling

Band structure of twisted bilayer graphene

Band structure of monolayer NbSe2

Chern number of an artificial Chern insulator

Topological phase transition in an artificial topological superconductor

Spatial distribution of Majorana modes in an artificial topological superconductor

Unfolded electronic structure of a supercell with a defect

Moire band structure of a moire superlattice

Unfolded electronic structure of a moire superstructure

Band structure of a nodal line semimetal slab

Band structure of a three dimensional topological insulator slab

Surface spectral function of a 2D quantum spin-Hall insulator

Local density of states with atomic orbitals of a honeycomb nanoisland

Interaction-driven magnetism in a honeycomb nanoisland

Hofstadter's butterfly of a square lattice

Landau levels of a Dirac semimetal

Surface spectral function of a Chern insulator

Surface states and Berry curvature of a artificial 2D topological superconductor

Surface states in a topological superconductor nanoisland

Antiferromagnet-superconductor interface

Fermi surface of a triangular lattice supercell

Unfolded Fermi surface of a supercell with a defect

Tunneling and Andreev reflection in a metal-superconductor junction

From tunneling to contact transport of a topological superconductor

Single impurity in an infinite honeycomb lattice

Bound state of a single magnetic impurity in an infinite superconductor

Parity switching of a magnetic impurity in an infinite superconductor

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