# How to tell if a state written in second quantization is a Slater determinant?

How do I tell if a state written in second quantization is a Slater determinant? I was solving some basic quantum-many body systems, and, for numerical purposes, I would like to determine if the ground state that I found is a Slater determinant.

• Commented Jul 29 at 7:52

Let $$\mathfrak h$$ denote a single-particle Hilbert space, and $$H_N:=\wedge^N \mathfrak h$$ the Hilbert space of $$N$$ identical fermions. We call a normalized vector $$\psi \in H_N$$ a Slater determinant if there exists $$\varphi_1,\varphi_2,\ldots,\varphi_N \in\mathfrak h$$ with $$\langle \varphi_i,\varphi_j\rangle_\mathfrak h=\delta_{ij}$$ for $$i,j\in \{1,2,\ldots, N\}$$ such that

$$\psi = \varphi_1\wedge \varphi_2\wedge \ldots \wedge \varphi_N = \frac{1}{\sqrt N} a^\dagger (\varphi_1)a^\dagger(\varphi_2)\cdots a^\dagger(\varphi_N)\, |0\rangle \quad . \tag 1$$

Given any $$\psi\in H_N$$, one can define an operator $$\gamma_\psi:\mathfrak h\rightarrow \mathfrak h$$, the so-called one-body reduced density matrix, via $$\langle \varphi, \gamma_\psi \phi\rangle_\mathfrak h := \langle \psi, a^\dagger(\phi)a(\varphi)\psi\rangle_{H_N} \quad . \tag 2$$

Theorem: A normalized vector $$\psi \in H_N$$ is a Slater determinant if and only if $$\gamma_\psi^2=\gamma_\psi$$.

The proof is by direct computation and can be found in e.g. Many-Particle Theory. E. Gross and E. Runge. Chapter 6, p. 49.

Thus, given your (normalized) ground state $$\psi$$, you can choose a suitable orthonormal single-particle basis $$\{e_j\}_{j=1,2,\ldots, \dim \mathfrak h}$$, construct the matrix $$(\gamma_\psi)_{ij}:=\langle \psi, a^\dagger(e_j)a(e_i)\psi\rangle_{H_N}$$ and simply check whether it is idempotent.

That being said, most interacting systems do not admit Slater determinant ground states. A notable exception is e.g. the Hubbard model with $$t=0$$, where at least one ground state is a Slater determinant.

• This time you succeeded to answer before me! Commented Jul 28 at 15:30

A many-fermion state that can be written as a single Slater determinant if and only if it satisfies the Plucker relations.