Tuto 5: Hybrid Fermion-Boson Systems
Lucie Pepe - Laboratoire de Chimie Quantique de Strasbourg, France - January 2024
Hybrid quantum many-body systems are prevalent in nature and they originate from the intricate interplay between electrons and bosons. These systems manifest, for example, when a molecular systems (or materials) interact with an external environment, incorporating photons and/or phonons. This is the case for exemple in polaritonic quantum chemistry, where the electronic structure of a molecule interacts with the photonic field of a cavity. Similarly, in condensed matter physics, the electronic degrees of freedom (or excitonic ones) experience perturbation owing to the presence of a vibrational phononic environment. From a theoretical point of view, the total number of electrons \(N_{elec}\) in these systems is conserved as constant, but this is not the case for the bosonic number \(N_{bos}\).
The QuantNBody package provides all the necessary tools to simulate such hybrid quantum many-body systems. In the subsequent sections, we will provide a comprehensive, step-by-step breakdown of its functionality. Before starting, let us import the package ;-)
# Import the quantnbody package
import quantnbody as qnb
import numpy as np
Step 1: Building a hybrid many-body basis
Hybrid quantum many-body systems can formally be described by a hybrid Hilbert space \(\mathcal{H}_{hyb}\) composed of a set of electron-boson configurations \(\lbrace | \Phi_{hyb}^\kappa\rangle \rbrace_{\kappa=1}^{ \dim(\mathcal{H}_{hyb})}\) expressed as:
Here, the bosonic configuration \(| \Psi_{bos} \rangle\) describes how \(N_{bos}\) bosons occupy \(N_{mode}\) modes, while the electronic configuration \(| \Psi_{elec} \rangle\) is a Slater determinant describing how \(N_{elec}\) electrons occupy a collection of \(2 N_{MO}\) spin-orbitals (i.e. \(N_{MO}\) spatial orbitals).
The total number of accessible fermion-boson configurations \(\lbrace | \Phi_{hyb}^\kappa\rangle \rbrace_{\kappa=1}^{ \dim(\mathcal{H}_{hyb})}\) is given by the dimension of the hybrid Hilbert space:
with
\(\dim({\mathcal{H}_{bos}})\) describes the dimension of the bosonic Fock-space. It includes all the possible distributions of \(N_{bos}\) bosons in \(N_{modes}\) modes (with \(N_b=0 \rightarrow N_{b}^{max}\)). Note again that this space doesn’t preserve the total number of bosons!
\(\dim({\mathcal{H}_{elec}})\) describes the electronic space counting all the possibilites to distribute \(N_{elec}\) in \(N_{MO}\) spatial orbitals.
How to create such a many-body basis with QuantNBody ?
The QuantNBody package builds a numerical representation of such a hybrid many-body basis as a list of states describing the repartition of \(N_{elec}\) electrons in \(2N_{MO}\) spin-orbitals, combined with the repartition of \(N_{b}\) bosons in \(N_{modes}\) modes. These states are numerically referenced by a list of kappa indices such that :
A little example with \(N_{MO}=N_{elec}=2\) and \(N_{bos}=N_{modes}=2\) : in this case, we should have 36 many-body states
# ======================================
# Define the fermionic system
# ======================================
N_elec = 2 # number of fermions
N_MO = 2 # number of molecular orbitals
# ======================================
# Define the bosonic system
# ======================================
N_b_max = 2 # maximal number of bosons in the whole system
N_mode = 2 # number of bosonic modes
list_bosons = range(N_b_max+1) # list of all possible number of bosons that can be distributed in the bosonic modes
# ======================================
# Build the hybrid many-body basis
# ======================================
nbody_basis = qnb.hybrid_fermionic_bosonic.tools.build_nbody_basis(N_mode, list_bosons, N_MO, N_elec)
# Print results
print('Shape of the hybrid kappa states')
for s in range(len(nbody_basis)):
print('| kappa={} >'.format(s), '=', nbody_basis[s])
Shape of the hybrid kappa states
| kappa=0 > = [0 0 1 1 0 0]
| kappa=1 > = [0 0 1 0 1 0]
| kappa=2 > = [0 0 1 0 0 1]
| kappa=3 > = [0 0 0 1 1 0]
| kappa=4 > = [0 0 0 1 0 1]
| kappa=5 > = [0 0 0 0 1 1]
| kappa=6 > = [1 0 1 1 0 0]
| kappa=7 > = [1 0 1 0 1 0]
| kappa=8 > = [1 0 1 0 0 1]
| kappa=9 > = [1 0 0 1 1 0]
| kappa=10 > = [1 0 0 1 0 1]
| kappa=11 > = [1 0 0 0 1 1]
| kappa=12 > = [0 1 1 1 0 0]
| kappa=13 > = [0 1 1 0 1 0]
| kappa=14 > = [0 1 1 0 0 1]
| kappa=15 > = [0 1 0 1 1 0]
| kappa=16 > = [0 1 0 1 0 1]
| kappa=17 > = [0 1 0 0 1 1]
| kappa=18 > = [2 0 1 1 0 0]
| kappa=19 > = [2 0 1 0 1 0]
| kappa=20 > = [2 0 1 0 0 1]
| kappa=21 > = [2 0 0 1 1 0]
| kappa=22 > = [2 0 0 1 0 1]
| kappa=23 > = [2 0 0 0 1 1]
| kappa=24 > = [1 1 1 1 0 0]
| kappa=25 > = [1 1 1 0 1 0]
| kappa=26 > = [1 1 1 0 0 1]
| kappa=27 > = [1 1 0 1 1 0]
| kappa=28 > = [1 1 0 1 0 1]
| kappa=29 > = [1 1 0 0 1 1]
| kappa=30 > = [0 2 1 1 0 0]
| kappa=31 > = [0 2 1 0 1 0]
| kappa=32 > = [0 2 1 0 0 1]
| kappa=33 > = [0 2 0 1 1 0]
| kappa=34 > = [0 2 0 1 0 1]
| kappa=35 > = [0 2 0 0 1 1]
What is the meaning of these 36 states ?
Here, each list of number string represents an hybrid many-body occupation number state. As an example, let’s check the first state for which we have :
Here we choose to structure the occupation numbers as follows:
Bosonic modes are expressed at the beginning of the list of numbers. Each value refers to the number of bosons in the associated bosonic mode.
For the following fermionic part, each couple of terms refer to a same spatial orbital, with an alternation of \(\alpha\) and \(\beta\) -spinorbitals.
Considering the 36 states contained in the list, we see that the first set of 6 lines contains all the possible fermionic configurations, for a vacuum bosonic configuration. Then, the next following states describe all possible fermionic configurations, for another bosonic configuration where we consider 1 boson in the two modes, and so on… until all the possible repartitions of the \(N_{bos}=0 \rightarrow N_{bos}^{max}\) bosons in the \(N_{mode}\) modes have been scanned.
Step 2: About building operators in the hybrid many-body basis
In QuantNbody, to treat hybrid systems two types of many-body operators are considered as elementary bricks:
Fermionic hopping operators \(\hat{a}^\dagger \hat{a}\) : indeed, every operator conserving the total number of fermionic particles can be decomposed as a serie of electronic hopping operators \(\hat{a}^\dagger \hat{a} \otimes \mathbb{1}_{bos}\) which only act on the fermionic part leave unchanged the bosonic part of the hybrid-states.
Bosonic creation/anihilation \(\hat{b}^\dagger/\hat{b}\) : indeed, every operator not conserving the total number of bosonic particles may be expressed as a series of anihilation (or creation) bosonic operators \(\hat{b} \otimes \mathbb{1}_{elec}\) that only act on the bosonic part of the states and leave the fermionic part unchanged.
The QuantNbody package provides a matrix representation of these two central operators in the numerical hybrid-many body basis. If we generically call the latter operators by \(\hat{O}\), this means in practice that we create a matrix representation such that
Step 3: Build the fermionic \(\hat{a}^\dagger \hat{a}\) operator in the hybrid basis
In the hybrid part of the QuanNBody package, the fermionic operators \(\hat{a}^\dagger_{p,\sigma} \hat{a}_{q,\tau}\) can be built with a simple command line:
a_dagger_a = qnb.hybrid_fermionic_bosonic.tools.build_fermion_operator_a_dagger_a(nbody_basis, N_mode)
In practice, the way “a_dagger_a” will store each operator \(\hat{a}^\dagger_{p,\sigma} \hat{a}_{q,\tau}\) follows the way we order the spin-orbitals in our many-body states. As an illustrative example, taking the following elements will return the associated many-body operators:
a_dagger_a[0,0] \(\longrightarrow \hat{a}^\dagger_{0,\alpha} \hat{a}_{0,\alpha}\)
a_dagger_a[1,0] \(\longrightarrow \hat{a}^\dagger_{0,\beta} \hat{a}_{0,\alpha}\)
a_dagger_a[10,1] \(\longrightarrow \hat{a}^\dagger_{5,\alpha} \hat{a}_{0,\beta}\)
Example of the matrix shape of the hopping operator \(\hat{a}^\dagger_{0,\alpha} \hat{a}_{1,\alpha}\) :
If we look at the element a_dagger_a[0,2], we get access to a sparse matrix representation of the fermionic operator \(\hat{a}^\dagger_{0,\alpha} \hat{a}_{1,\alpha}\) in the hybrid many-body basis which encodes the promotion of 1 electron from the the 2nd spin orbital (second MO, spin up) to the 0th spin orbital (first MO, spin up) of the fermionic sub-system:
print(a_dagger_a[0,2])
(0, 3) -1.0
(2, 5) 1.0
(6, 9) -1.0
(8, 11) 1.0
(12, 15) -1.0
(14, 17) 1.0
(18, 21) -1.0
(20, 23) 1.0
(24, 27) -1.0
(26, 29) 1.0
(30, 33) -1.0
(32, 35) 1.0
We observe here that the action of this operator is only possible between specific configurations. As an exemple, let us consider the first line that shows a connexion between the \(\kappa\) states \(|0 \rangle \leftrightarrow | 3 \rangle\). These two states are actually given by | kappa=0 > = [0 0 1 1 0 0] and | kappa=3 > = [0 0 0 1 1 0]. Here, we clearly see that the action of the operator is well encoded as it generates the following modification:
The electron hops between the 0th and the 2nd spin-orbitals.
There is no change in the occupation number of the bosonic modes between these two states.
Step 4: Build the bosonic \(\hat{b}\) and \(\hat{b}^\dagger\) operators in the hybrid basis
In the hybrid part of the QuanNBody package, the bosonic anihilation operators \(\hat{b}_p\) can be built with a simple command line:
# We compute here the b operator
b = qnb.hybrid_fermionic_bosonic.tools.build_boson_anihilation_operator_b(nbody_basis,N_mode)
In practice, the way “b” will store each operator \(\hat{b}_{p}\) follows the way we order the modes in our many-body states. As an illustrative example, taking the following elements will return the associated many-body operators:
b[p] \(\longrightarrow \hat{b}_{p}\)
Note that we can easily build the associated creation operator by taking the tranposed version of each element such that
b[p].T \(\longrightarrow \hat{b}_{p}^\dagger\)
Example of a bosonic anihilation operator \(\hat{b}_0\) :
If we look at the element b[0], we get access to a sparse matrix representation of the bosonic anihilation operator \(\hat{b}_0\) in the hybrid many-body basis which encodes the desctruction of 1 boson in the 0th mode:
print(b[0])
(0, 6) 1.0
(1, 7) 1.0
(2, 8) 1.0
(3, 9) 1.0
(4, 10) 1.0
(5, 11) 1.0
(6, 18) 1.4142135623730951
(7, 19) 1.4142135623730951
(8, 20) 1.4142135623730951
(9, 21) 1.4142135623730951
(10, 22) 1.4142135623730951
(11, 23) 1.4142135623730951
(12, 24) 1.0
(13, 25) 1.0
(14, 26) 1.0
(15, 27) 1.0
(16, 28) 1.0
(17, 29) 1.0
We observe here that the action of this operator is only possible between specific configurations. As an exemple, let us consider the first element that shows a connexion between the \(\kappa\) states \(|0 \rangle \leftrightarrow | 6 \rangle\). These two states are actually given by | kappa=0 > = [0 0 1 1 0 0] and | kappa=6 > = [1 0 1 1 0 0]. Here, we clearly see that the action of the operator is well encoded:
The two states are related by the creation/anhihilation of one boson in the 0th mode.
There is no change in the fermionic occupation numbers of the spin-orbitals between the two states.
Last exemple with a counting \(\hat{b}_1^\dagger\hat{b}_1\) operator:
Once all the \(\hat{b}_p\) are built, one can use these operators as building blocks for a wide possibilty of operators such as the \(\hat{n}_p = \hat{b}^\dagger_p \hat{b}_p\) counting one. As an exemple, let’s count the number of bosons in the second mode of the following state we want to target
QuantNBody provides a way to build our own state from a given occupation number list as follows
# 1) Define the occupation number list of bosonic modes and fermionic spin-orbitals
LIST_OCC_NUMB = [0,2,1,1,0,0]
# 2) Obtain the qnb traduction in the hybrid many-body basis
my_many_body_state = qnb.hybrid_fermionic_bosonic.tools.my_state(LIST_OCC_NUMB, nbody_basis)
# 2) Visualize the associated wavefunction
print( 'Initial state :')
qnb.hybrid_fermionic_bosonic.tools.visualize_wft(my_many_body_state,
nbody_basis,
N_mode )
print()
Initial state :
-----------
Coeff. N-body state and index
------- ----------------------
+1.00000 |02⟩ ⊗ |1100⟩ #30
Let us now count the number of bosons in this state as follows:
n_1 = b[1].T@b[1]
print("Total number of boson in the targeted state\n", my_many_body_state.T @ n_1 @ my_many_body_state )
Total number of boson in the targeted state
2.0000000000000004