# 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
Jul 3 '19 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
Jul 3 '19 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
Jul 3 '19 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. Jul 3 '19 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". Jul 3 '19 at 19:19