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!

  • $\begingroup$ Thanks for your comments. To prevent 2 fermion species to be at the same site-can't we encode this using commutators alone? $\endgroup$
    – jcp
    Jul 3 '19 at 18:56
  • $\begingroup$ 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. $\endgroup$
    – jcp
    Jul 3 '19 at 18:57
  • $\begingroup$ As far as occupation number of the vacancies is concerned then, should a vacancy be treated as a particle as well then? $\endgroup$
    – jcp
    Jul 3 '19 at 19:00
  • $\begingroup$ @jcp, non-hermitian hamiltonians are usually associated with dissipative systems. To make your Hamiltonian hermitian, you may include the hermitian-conjugate (h.c.) counterpart. $\endgroup$
    – MadMax
    Jul 3 '19 at 19:09
  • $\begingroup$ @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". $\endgroup$
    – MadMax
    Jul 3 '19 at 19:19

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