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partyday
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- TL;DR Summary
- I am asking about how to convert the Hard-Core boson Model into K-space.
Hello,
I am interested in the following model:
$$
H = \sum_{<i,j>} -t (c_i c_j^{\dagger} + \text{H.C.}) + U (n_i n_j) + \sum_{<<i,j>>} -t' (c_i c_j^{\dagger} + \text{H.C.}) + U' (n_i n_j)
$$
where \( <i,j> \) indicates nearest neighbors, and \( <<i,j>> \) indicates next-nearest neighbors interactions. \( c_i \) and its conjugate are the annihilation and creation operators, respectively.
The hardcore boson model also establishes that
$$
c_i^{\dagger} c_i^{\dagger} = c_i c_i = 0
$$
when acting on a state, so a particle may only inhabit one site at a time.
If we establish that there are \( L \) sites on a periodic lattice (so \( i+L = i \)), then this Hamiltonian can be rewritten:
$$
H = \sum_{i} -t (c_i c_{i+1}^{\dagger} + \text{H.C.}) + U (n_i n_{i+1}) - t' (c_i c_{i+2}^{\dagger} + \text{H.C.}) + U' (n_i n_{i+2})
$$
This Hamiltonian can be block diagonalized into different sectors of momenta, \( K \). For this reason, I'd like to express this Hamiltonian in terms of momentum states.
Using
$$
\omega = \frac{2 \pi}{L}
$$
and
$$
c_j = \sum_{k} c_k e^{-i \omega k j}
$$
we can derive the following two expressions:
$$
\sum_{j} n_j n_{j+x} = \sum_{j} \sum_{k, k', q, q'} c^{\dagger}_{k'} c^{\dagger}_{q'} c_k c_q e^{i (k'+q'-k-q) j \omega} e^{i (q'-q) x \omega}
$$
which simplifies to
$$
\sum_{j} n_j n_{j+x} = \sum_{k, k', q, q'} \delta(k'+q'-k-q) e^{i (q'-q) x \omega} c^{\dagger}_{k'} c^{\dagger}_{q'} c_k c_q
$$
The diagonal elements are more simple:
$$
\sum_j c^{\dagger}_{j+x} c_j + \text{H.C.} = \sum_k 2 \cos(k \omega x) n_k
$$
Now these terms can be substituted into the Hamiltonian for \( x = 1 \), \( x = 2 \).
My questions are as follows:
1. Was this derivation correct?
2. In momentum space, does
$$
c_k^{\dagger} c_k^{\dagger} = c_k c_k = 0
$$
hold?
3. Should values of $$ K $$ vary from $$( (-L/2, -L/2+1, \ldots, L/2-1, L/2) $$ or from $$ (0, 1, \ldots, L-1) $$? Is there a difference?
4. I'm interested in this model when there is an appreciable amount of filling. Say $$N = L/3 $$. I have been attempting to program a script that computes the matrix elements of this Hamiltonian for an arbitrary $$L $$ and $$ N $$. I first find all available basis states in momentum space, then I separate them into total momentum sectors. Say three particles with momenta $$ (1, 2, 5) $$ are in the same sector as $$ (1, 3, 4) $$ as they sum to the same number $$ K = 8$$. The action of the $$ n_j n_{j+x}$$ term in momentum space is to couple these states together (as well as any other states that are reachable via a momentum-conserving interaction). But let's say $$ L = 6 $$, then these two states are of $$K \% L $$, correct? It would be correct to say that these are of momentum $$ K = 2 $$, because it must lie in the first Brillouin zone?
5. Are these $$ K $$ sectors block diagonalizable themselves? Some papers I've seen make reference to parity blocks, but none define what this would look like. I can see, for instance, very clearly that $$ (1, 3, 4)$$ and $$(1, 2, 5)$$ are in the same block of total momentum $ K $, but how can I see if they are in the same parity block?
Any help or guidance would be greatly appreciated. I have tried a lot of academic papers and Google-searching but I have not been able to feel assured in my understanding yet.
I am interested in the following model:
$$
H = \sum_{<i,j>} -t (c_i c_j^{\dagger} + \text{H.C.}) + U (n_i n_j) + \sum_{<<i,j>>} -t' (c_i c_j^{\dagger} + \text{H.C.}) + U' (n_i n_j)
$$
where \( <i,j> \) indicates nearest neighbors, and \( <<i,j>> \) indicates next-nearest neighbors interactions. \( c_i \) and its conjugate are the annihilation and creation operators, respectively.
The hardcore boson model also establishes that
$$
c_i^{\dagger} c_i^{\dagger} = c_i c_i = 0
$$
when acting on a state, so a particle may only inhabit one site at a time.
If we establish that there are \( L \) sites on a periodic lattice (so \( i+L = i \)), then this Hamiltonian can be rewritten:
$$
H = \sum_{i} -t (c_i c_{i+1}^{\dagger} + \text{H.C.}) + U (n_i n_{i+1}) - t' (c_i c_{i+2}^{\dagger} + \text{H.C.}) + U' (n_i n_{i+2})
$$
This Hamiltonian can be block diagonalized into different sectors of momenta, \( K \). For this reason, I'd like to express this Hamiltonian in terms of momentum states.
Using
$$
\omega = \frac{2 \pi}{L}
$$
and
$$
c_j = \sum_{k} c_k e^{-i \omega k j}
$$
we can derive the following two expressions:
$$
\sum_{j} n_j n_{j+x} = \sum_{j} \sum_{k, k', q, q'} c^{\dagger}_{k'} c^{\dagger}_{q'} c_k c_q e^{i (k'+q'-k-q) j \omega} e^{i (q'-q) x \omega}
$$
which simplifies to
$$
\sum_{j} n_j n_{j+x} = \sum_{k, k', q, q'} \delta(k'+q'-k-q) e^{i (q'-q) x \omega} c^{\dagger}_{k'} c^{\dagger}_{q'} c_k c_q
$$
The diagonal elements are more simple:
$$
\sum_j c^{\dagger}_{j+x} c_j + \text{H.C.} = \sum_k 2 \cos(k \omega x) n_k
$$
Now these terms can be substituted into the Hamiltonian for \( x = 1 \), \( x = 2 \).
My questions are as follows:
1. Was this derivation correct?
2. In momentum space, does
$$
c_k^{\dagger} c_k^{\dagger} = c_k c_k = 0
$$
hold?
3. Should values of $$ K $$ vary from $$( (-L/2, -L/2+1, \ldots, L/2-1, L/2) $$ or from $$ (0, 1, \ldots, L-1) $$? Is there a difference?
4. I'm interested in this model when there is an appreciable amount of filling. Say $$N = L/3 $$. I have been attempting to program a script that computes the matrix elements of this Hamiltonian for an arbitrary $$L $$ and $$ N $$. I first find all available basis states in momentum space, then I separate them into total momentum sectors. Say three particles with momenta $$ (1, 2, 5) $$ are in the same sector as $$ (1, 3, 4) $$ as they sum to the same number $$ K = 8$$. The action of the $$ n_j n_{j+x}$$ term in momentum space is to couple these states together (as well as any other states that are reachable via a momentum-conserving interaction). But let's say $$ L = 6 $$, then these two states are of $$K \% L $$, correct? It would be correct to say that these are of momentum $$ K = 2 $$, because it must lie in the first Brillouin zone?
5. Are these $$ K $$ sectors block diagonalizable themselves? Some papers I've seen make reference to parity blocks, but none define what this would look like. I can see, for instance, very clearly that $$ (1, 3, 4)$$ and $$(1, 2, 5)$$ are in the same block of total momentum $ K $, but how can I see if they are in the same parity block?
Any help or guidance would be greatly appreciated. I have tried a lot of academic papers and Google-searching but I have not been able to feel assured in my understanding yet.
Last edited:
