FreeFermionPoint

The free fermion point in a quantum integrable model refers to a specific point or set of conditions in the parameter space of the model where the system's behavior can be exactly described by non-interacting fermions. This concept is often discussed in the context of exactly solvable models in statistical mechanics and quantum field theory.

To explain this with equations, let's consider a basic example from the realm of integrable models: the 1D Ising model in a transverse field. The Hamiltonian for this model is given by:

$$ H = -J \sum_{i} \sigma_i^z \sigma_{i+1}^z - h \sum_{i} \sigma_i^x $$

where $\sigma_i^z$ and $\sigma_i^x$ are the Pauli matrices at site $i$, $J$ is the coupling constant, and $h$ is the transverse field. This model is exactly solvable using the Jordan-Wigner transformation, which maps the spin operators to fermionic creation and annihilation operators. After this transformation, the Hamiltonian becomes:

$$ H = -J \sum_{i} (c_i^\dagger c_{i+1} + c_{i+1}^\dagger c_i) - 2h \sum_{i} (c_i^\dagger c_i - \frac{1}{2}) $$

where $c_i^\dagger$ and $c_i$ are the fermionic creation and annihilation operators, respectively.

The free fermion point occurs at $J = h$. At this point, the model can be solved by considering non-interacting fermions, and the Hamiltonian simplifies to a form that describes a system of free fermions. The energy spectrum and the correlation functions can be computed exactly, providing deep insights into the physics of the model near this point.

This is just one example, and the concept of a free fermion point can be extended to other integrable models as well. The key aspect of the free fermion point is that the complexity of interactions in the model reduces to a level where it can be mapped to a system of non-interacting fermions, making analytical solutions possible.