# Second quantisation for fermions

I am trying to build a model for reactions on a lattice in the Doi-Peliti formalism. Suppose there exists a lattice of $$N$$ sites indexed by $$i$$. Each site can be either occupied or unoccupied. Assuming there exists a single type of particle, I can use $$SU(2)$$ fermionic operators: $$a^\dagger$$ and $$a$$ to denote creation and annihilation operators that obey the anti-commutation rules: (subscript indicates lattice site) $$\{a_i,a^\dagger_j\} = \delta_{i,j}$$ $$\{a_i,a_j\}= \{a^\dagger_i, a^\dagger_j\} = 0$$

Now suppose there is more than one type of fermion (say $$a^{(1)}$$ and $$a^{(2)}$$), however, each lattice site can either be unoccupied or be occupied by either exactly one $$a^{(1)}$$ or $$a^{(2)}$$ but not both.

First question, what would be the appropriate commutation rules in this case,

I assume the following are still valid: $$\{a^{(x)}_i,a^{(x)\dagger}_i\} = 1$$ $$[a^{(x)}_i,a^{(y)\dagger}_j] = 0 \qquad \text{if} x \neq y\ \text{and}\ i \neq j$$

However, what about $$[a^{(x)}_i,a^{(y)\dagger}_i] = ? \qquad \text{if} x \neq y\ \text{and}\ i = j$$

Again, I want each site to be only singly occupied (either by $$a^{(1)}$$ or $$a^{(2)}$$) or unoccupied.

Second, would these commutators be enough to characterise the system or do I need something more?

Third, am I correct to assume that the number operators for $$a^{(1)}$$, $$a^{(2)}$$ and vacancies would be given by $$N_i^{(1)} = a^{(1)\dagger}_ia^{(1)}_i$$ $$N_i^{(2)} = a^{(2)\dagger}_ia^{(2)}_i$$ and $$N_i^{(\text{vac})} = 1 - N_i^{(1)}- N_i^{(2)}$$

I suspect this problem might be vaguely connected to parastatistics and Green ansatz, but I am not certain.

Fourth, now in Doi-Peliti formalism a reaction where particle at site $$i$$ interacts with its neighbour at $$j$$ and is turned to C: $$A_i + B_j \rightarrow C_i+ B_j$$ would be given by the hamiltonian: ($$j(i)$$ indicates summing over sites neighbouring $$i$$). Typically I am familiar with the situation of unrestricted occupation numbers where the operators are bosonic, however would this still hold in the case of restricted occupation numbers using fermionic operators described above.

$$H = k \sum_{j(i)}b^{\dagger}_jb_j(c^\dagger_ia_i-a^\dagger_ia_i)$$ Now, consider the case wherein a vacancy is created instead of a new particle.

$$A_i + B_j \rightarrow \emptyset + B_j$$

Should the vacancy be treated just like a particle in this case? Or is the hamiltonian simply

$$H = k \sum_{j(i)} b^{\dagger}b_j(a_i-a^\dagger_ia_i)$$

$$[a^{(x)}_i,a^{(y)\dagger}_i] = ? \qquad \text{if} x \neq y\ \text{and}\ i = j$$

$$[a^{(x)}_i,a^{(y)\dagger}_i] = 0 \qquad \text{if} x \neq y\ \text{and}\ i = j$$. Any two fermion species anti-commute, regardless of them being Jigglypuff ,Meowth, or what have ya.

would these commutators be enough to characterise the system or do I need something more?

To characterize the system, you need the Hamiltonian/Lagrangian, as well as initial condition or density matrix.

vacancies would be given by ... $$N_i^{(\text{vac})} = 1 - N_i^{(1)}- N_i^{(2)}$$

If both species (1 and 2) were at site $$i$$ (nothing preventing them being at the same site), your vacancies operator would yield -1.

I want each site to be only singly occupied.

Try adding a term like $$H_{int} = U \sum_{i} b^{\dagger}_ib_ia^\dagger_ia_i$$ with a large $$U$$ to make double occupancy energetically costly.

$$H = k \sum_{j(i)} b^{\dagger}b_j(a_i-a^\dagger_ia_i)$$

Your Hamiltonian does not past the litmus test of being Hermitian!

• Thanks for your comments. To prevent 2 fermion species to be at the same site-can't we encode this using commutators alone?
– jcp
Commented Jul 3, 2019 at 18:56
• Well, the way I understand it is in the Doi-Peliti formalism the hamiltonian need not be hermitian. One of the references I am following is this (arxiv.org/pdf/1209.3632.pdf). Unfortunately, it doesn't spend much time on fermionic operators.
– jcp
Commented Jul 3, 2019 at 18:57
• As far as occupation number of the vacancies is concerned then, should a vacancy be treated as a particle as well then?
– jcp
Commented Jul 3, 2019 at 19:00
• @jcp, non-hermitian hamiltonians are usually associated with dissipative systems. To make your Hamiltonian hermitian, you may include the hermitian-conjugate (h.c.) counterpart. Commented Jul 3, 2019 at 19:09
• @jcp, to prevent double occupancy, some would add a punitive term (Hubbard term, as suggested in my answer), whereby the weight of doubly occupied orbitals is reduced in the ground state via "Gutzwiller projection". Commented Jul 3, 2019 at 19:19