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Add two-dimensional Fermi-Hubbard model #141

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2 changes: 2 additions & 0 deletions python/ffsim/__init__.py
Original file line number Diff line number Diff line change
Expand Up @@ -42,6 +42,7 @@
des_a,
des_b,
fermi_hubbard_1d,
fermi_hubbard_2d,
number_operator,
)
from ffsim.protocols import (
Expand Down Expand Up @@ -126,6 +127,7 @@
"expectation_one_body_power",
"expectation_one_body_product",
"fermi_hubbard_1d",
"fermi_hubbard_2d",
"fermion_operator",
"hartree_fock_state",
"indices_to_strings",
Expand Down
3 changes: 2 additions & 1 deletion python/ffsim/operators/__init__.py
Original file line number Diff line number Diff line change
Expand Up @@ -11,7 +11,7 @@
"""Operators."""

from ffsim.operators.common_operators import number_operator
from ffsim.operators.fermi_hubbard import fermi_hubbard_1d
from ffsim.operators.fermi_hubbard import fermi_hubbard_1d, fermi_hubbard_2d
from ffsim.operators.fermion_action import (
FermionAction,
cre,
Expand All @@ -33,5 +33,6 @@
"des_a",
"des_b",
"fermi_hubbard_1d",
"fermi_hubbard_2d",
"number_operator",
]
108 changes: 108 additions & 0 deletions python/ffsim/operators/fermi_hubbard.py
Original file line number Diff line number Diff line change
Expand Up @@ -10,6 +10,8 @@

"""Fermi-Hubbard model Hamiltonian."""

from collections import defaultdict

from ffsim._lib import FermionOperator
from ffsim.operators.fermion_action import cre_a, cre_b, des_a, des_b

Expand Down Expand Up @@ -85,3 +87,109 @@ def fermi_hubbard_1d(
coeffs[(cre_b(p), des_b(p))] = -chemical_potential

return FermionOperator(coeffs)


def fermi_hubbard_2d(
norb_x: int,
norb_y: int,
tunneling: float,
interaction: float,
*,
chemical_potential: float = 0,
nearest_neighbor_interaction: float = 0,
periodic: bool = False,
) -> FermionOperator:
r"""Two-dimensional Fermi-Hubbard model Hamiltonian.

The Hamiltonian for the two-dimensional Fermi-Hubbard model on a square lattice with
:math:`N = N_x \times N_y` spatial orbitals is given by

.. math::

H = -t \sum_{\sigma} \sum_{\braket{pq}}
(a^\dagger_{\sigma, p} a_{\sigma, q} + \text{h.c.})
+ U \sum_{p=1}^{N} n_{\alpha, p} n_{\beta, p}
- \mu \sum_{p=1}^{N} (n_{\alpha, p} + n_{\beta, p})
+ V \sum_{\sigma, \sigma'} \sum_{\braket{pq}} n_{\sigma, p} n_{\sigma', q}

where :math:`\braket{\dots}` denotes nearest-neighbor pairs and
:math:`n_{\sigma, p} = a_{\sigma, p}^\dagger a_{\sigma, p}` is the number operator
on orbital :math:`p` with spin :math:`\sigma`.

References:
- `The Hubbard Model`_

Args:
norb_x: The number of spatial orbitals in the x-direction :math:`N_x`.
norb_y: The number of spatial orbitals in the y-direction :math:`N_y`.
tunneling: The tunneling amplitude :math:`t`.
interaction: The onsite interaction strength :math:`U`.
chemical_potential: The chemical potential :math:`\mu`.
nearest_neighbor_interaction: The nearest-neighbor interaction strength
:math:`V`.
periodic: Whether to use periodic boundary conditions.

Returns:
The two-dimensional Fermi-Hubbard model Hamiltonian.

.. _The Hubbard Model: https://doi.org/10.1146/annurev-conmatphys-031620-102024
"""
coeffs: dict[tuple[tuple[bool, bool, int], ...], complex] = defaultdict(float)
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for x in range(norb_x):
for y in range(norb_y):
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# position in Cartesian coordinates
x_right = (x + 1) % norb_x
y_up = (y + 1) % norb_y

# position on C-style chain
p = norb_y * x + y
p_right = norb_y * x_right + y
p_up = norb_y * x + y_up

if x != norb_x - 1 or periodic:
coeffs[(cre_a(p), des_a(p_right))] -= tunneling
coeffs[(cre_a(p_right), des_a(p))] -= tunneling
coeffs[(cre_b(p), des_b(p_right))] -= tunneling
coeffs[(cre_b(p_right), des_b(p))] -= tunneling
if nearest_neighbor_interaction:
coeffs[(cre_a(p), des_a(p), cre_a(p_right), des_a(p_right))] = (
nearest_neighbor_interaction
)
coeffs[(cre_a(p), des_a(p), cre_b(p_right), des_b(p_right))] = (
nearest_neighbor_interaction
)
coeffs[(cre_b(p), des_b(p), cre_a(p_right), des_a(p_right))] = (
nearest_neighbor_interaction
)
coeffs[(cre_b(p), des_b(p), cre_b(p_right), des_b(p_right))] = (
nearest_neighbor_interaction
)

if y != norb_y - 1 or periodic:
coeffs[(cre_a(p), des_a(p_up))] -= tunneling
coeffs[(cre_a(p_up), des_a(p))] -= tunneling
coeffs[(cre_b(p), des_b(p_up))] -= tunneling
coeffs[(cre_b(p_up), des_b(p))] -= tunneling
if nearest_neighbor_interaction:
coeffs[(cre_a(p), des_a(p), cre_a(p_up), des_a(p_up))] = (
nearest_neighbor_interaction
)
coeffs[(cre_a(p), des_a(p), cre_b(p_up), des_b(p_up))] = (
nearest_neighbor_interaction
)
coeffs[(cre_b(p), des_b(p), cre_a(p_up), des_a(p_up))] = (
nearest_neighbor_interaction
)
coeffs[(cre_b(p), des_b(p), cre_b(p_up), des_b(p_up))] = (
nearest_neighbor_interaction
)

for p in range(norb_x * norb_y):
if interaction:
coeffs[(cre_a(p), des_a(p), cre_b(p), des_b(p))] = interaction
if chemical_potential:
coeffs[(cre_a(p), des_a(p))] = -chemical_potential
coeffs[(cre_b(p), des_b(p))] = -chemical_potential

return FermionOperator(coeffs)
2 changes: 1 addition & 1 deletion tests/python/operators/__init__.py
Original file line number Diff line number Diff line change
@@ -1,4 +1,4 @@
# (C) Copyright IBM 2023.
# (C) Copyright IBM 2024.
#
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# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
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