Fermion Operators

This page details the specific operators available for Fermionic systems. These operators implement the general Operator interface described in the Operators concept guide.

Fermionic Algebra and State Representation

Fermionic operators differ fundamentally from spin/qubit operators due to their anti-commutation relations. The creation ($f_i^\dagger$) and annihilation ($f_i$) operators satisfy:

$\{f_i, f_j^\dagger\} = f_i f_j^\dagger + f_j^\dagger f_i = \delta_{ij}$ $\{f_i, f_j\} = 0, \quad \{f_i^\dagger, f_j^\dagger\} = 0$

This implies the Pauli exclusion principle, where attempting to create two fermions on the same site yields zero ($f_i^\dagger f_i^\dagger = 0$).

When applied to basis states $|n\rangle$ where occupancy $n \in \{0, 1\}$:

• Annihilation: $f_i |1\rangle = |0\rangle$, but $f_i |0\rangle = 0$ (annihilating the vacuum "kills" the state).
• Creation: $f_i^\dagger |0\rangle = |1\rangle$, but $f_i^\dagger |1\rangle = 0$ (creating on an occupied site "kills" the state).

Fermionic operators in Aleph automatically handle the sign changes arising from these anti-commutation rules when permuting operators past each other.

Static Operators

Static operators define fixed physical actions such as creation, annihilation, hopping, and number interactions. They are purely symbolic and parametrized only by the site indices they act upon.

Factory Function	Sites	Description
Create(site)	1	Creation operator $f_i^\dagger$.
CreateCreate(i,j)	1	Product of two creation operators $f_i^\dagger f_j^\dagger$.
Destroy(site)	1	Annihilation operator $f_i$.
DestroyDestroy(i,j)	1	Product of two annihilation operators $f_i f_j$.
Number(site)	1	Number operator $n_i = f_i^\dagger f_i$.
NumberNumber(site1, site2)	2	Number interaction $n_i n_j$.
Hop(i, j)	2	Hermitian hopping $f_i^\dagger f_j + f_j^\dagger f_i$.
Transfer(dest, src)	2	Unidirectional transfer $f_{\text{dest}}^\dagger f_{\text{src}}$.
TransferNumber(dest, src, ctrl)	3	Conditional transfer $f_{\text{dest}}^\dagger f_{\text{src}} n_{\text{ctrl}}$.
HopNumber(i, j, ctrl)	3	Conditional hopping $(f_i^\dagger f_j + f_j^\dagger f_i) n_{\text{ctrl}}$.
NumberSum(sites)	arbitrary	Total occupancy on a set of sites $\sum_{i \in \mathrm{sites}} n_{i}$ (legacy: NumberN).
NearestNumberSum(sites)	arbitrary	Nearest-neighbor number interaction $\sum_{i} n_{i} n_{i+1}$ (legacy: NumberNumberNN).
NextNearestNumberSum(sites)	arbitrary	Next-nearest-neighbor number interaction $\sum_{i} n_{i} n_{i+2}$ (legacy: NumberNumberNNN).
PairTransfer(i, j, k, l)	4	Interaction term $f_i^\dagger f_j^\dagger f_k f_l$ (legacy: FourBody).
PairHop(i, j, k, l)	4	Double excitation $f_i^\dagger f_j^\dagger f_k f_l + h.c.$ (legacy: DoubleExcitation).

Parameterized Operators

Parameterized operators include continuous variables representing phases or rotation angles. These are often used in time-evolution or variational ansatzes.

Factory Function	Parameters	Sites	Description
PhaseHop(phi, i, j)	phi (real)	2	Complex hopping $e^{i\phi} f_i^\dagger f_j + e^{-i\phi} f_j^\dagger f_i$.
NumberPhase(phi, site)	phi (real)	1	Number phase $e^{i\phi n_i}$.
NumberNumberPhase(phi, i, j)	phi (real)	2	Interaction phase $e^{i\phi n_i n_j}$.
ModeRotation(theta, i, j)	theta (real)	2	Basis rotation $e^{i\theta (f_i^\dagger f_j + f_j^\dagger f_i)}$.
PairRotation(theta, i, j)	theta (real)	2	Pairing rotation $e^{i\theta (f_i^\dagger f_j^\dagger + f_j f_i)}$.
NumberPotential(coefficients [, sites])	coefficients (array/list)	arbitrary	Weighted number sums $\sum_{i} \alpha_i n_i$.

Tensor and Matrix Operators

For more complex interactions or efficient simulation of non-interacting systems, Aleph provides specialized operator types.

Type	Description
CoulombSum(tensor)	General Coulomb interaction $\sum_{i,j,k,l,\sigma,\sigma'} V_{ijkl} a_{i\sigma}^\dagger a_{j\sigma'}^\dagger a_{k\sigma'} a_{l\sigma}$ on interleaved spin-orbitals.
FreeOperatorMatrix	Specialized class for non-interacting (free) fermion Hamiltonians, allowing for highly efficient Gaussian state evolution.

Quadratic Operators

For the efficient handling of quadratic fermionic models, Aleph provides the specialized operator types TransferSum, PairingSum and QuadraticSum.

Factory Function	Parameters	Sites	Description
TransferSum(A)	A (matrix)	Contiguous [0,...,L-1] for a passed $L\times L$ matrix	Represents the sum $\sum_{j,k}A_{j,k}f_j^{\dagger}f_k$.
TransferSum(operator [, options])	operator Operator	Contiguous [0,...,max(operator.support())]	Efficiently represents the passed sum of quadratic particle number preserving fermionic terms. A scalar option can be provided.
PairingSum(P)	P (matrix)	Contiguous [0,...,L-1] for a passed $L\times L$ matrix	Represents the sum $\sum_{j < k}P_{j,k}f_j^\dagger f_k^\dagger + \sum_{j > k}P_{j,k}f_j f_k$
PairingSum(operator [, options])	operator Operator	Contiguous [0,...,max(operator.support())]	Efficiently represents the passed sum of quadratic fermionic pairing terms. A scalar option can be provided.
QuadraticSum(S)	S (matrix)	Contiguous [0,...,L-1] for a passed $L\times L$ matrix	Represents the sum $\sum_{j,k}S_{j,k}^{+,-}f_j^{\dagger}f_k + S_{j,k}^{-,+}f_jf_k^{\dagger} + S_{j,k}^{+,+}f_j^{\dagger}f_k^{\dagger} + S_{j,k}^{-,-}f_jf_k$ with $S =\begin{pmatrix}S^{+,-}, S^{+,+} \\ S^{-,-}, S^{-,+} \end{pmatrix}$
QuadraticSum(operator [, options])	operator Operator	Contiguous [0,...,max(operator.support())]	Efficiently represents the passed sum of quadratic fermionic terms. A scalar option can be provided.

Documentation Contributors

Vincent Michaud-Rioux

Pierre-Gabriel Rozon