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I want to exactly diagonalize the following Hamiltonian for $10$ number of sites and $4$ number of spinless fermions $$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{L-1} n_i n_{i+1}$$ here $L$ is total number of sites, creation ($c^\dagger$) and annihilation ($c$) operators are defined as following $$ c = \begin{bmatrix} 0&0\\1&0 \end{bmatrix} $$ and $n_i = c_i^\dagger c_i$ is number operator.

To exactly diagonalize (for simplicity let's take $L=4$ sites), one can expand $H$ as

$$H = -t\big[ c_1^\dagger \sigma_1^z \otimes c_2\otimes I_3 \otimes I_4 \\ + I_1 \otimes c_2^\dagger \sigma_2^z \otimes c_3\otimes I_4\\ + I_1 \otimes I_2 \otimes c_3^\dagger\sigma_3^z \otimes c_4 \big]+h.c.\\ +V\big[ n_1 \otimes n_2 \otimes I_3 \otimes I_4\\ +I_1 \otimes n_2 \otimes n_3 \otimes I_4\\ +I_1 \otimes I_2 \otimes n_3 \otimes n_4 \big]$$ where $\sigma^z$ (Pauli matrix) is just simple matrix multiplication for the sake of anti-commutation relation.

So far so good. (please correct me if I am doing anything wrong)!

Question:

I used the above method and numerically calculated the ground state and found that above method gives correct results for $V=0$ but when $V\ne0$ the results are wrong.

Eventually, I get to the point that I am not taking care of number of particles in the system. How do we numerically diagonalize a Hamiltonian matrix in the sector with chosen number of particles?

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  • $\begingroup$ Can you show that $c_1^\dagger$ and $c_2$ anti-commute? If I follow your recipe and try to calculate the anticommutator of $c_1^\dagger = c^\dagger\sigma^z \otimes I$ and $c_2 = I \otimes \sigma^z c$ then I do not get the null matrix. $\endgroup$
    – wcc
    Commented Jan 28, 2020 at 22:24

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If you want to numerically diagonalize a Hamiltonian, it is recommended to construct nonzero elements $H_{ij} = \langle i|H|j\rangle$ instead of using the Kronecker product $\otimes$. You can follow this algorithm:

  1. Build the Wannier basis.
  2. Apply the Hamiltonian to each basis state.
  3. Utilize algebra software to solve the eigenvalue problem.

Here's an example for a system with $L = 4$ sites and $N = 2$ particles:

  1. Generate the basis, which consists of all possible states for $L$ sites with $N$ particles: ${0: |0011\rangle, 1: |0101\rangle, 2: |1010\rangle, 3: |1100\rangle, 4: |1001\rangle, 5: |0110\rangle}$. To generate and enumerate combinations, you can use combinadics.
  2. Apply the Hamiltonian:
    • The interaction term $n_i n_j$ produces diagonal elements, for example: $H_{0,0} = \langle 0011|H|0011 \rangle = V$.
    • The hopping term $c_i^\dagger c_{i+1}$ produces off-diagonal elements, for example: $H_{5,2} = \langle0110|H|1010\rangle = -t$ (Note: Fermions anticommute, so in the hopping term, one should consider fermion operator permutations. If required, multiply this term by $-1$).
  3. Use appropriate software like Matlab, Armadillo (C++), or numpy (Python) to solve the eigenvalue problem for $H_{ij}$.
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    $\begingroup$ @CuriousMind Apologies for the delay. Note that in the case of bosonic operators, the task is much simpler since bosonic operators do commute. You have to enumerate your rows/cols using the states, e.g assign row: 10 and col 16 to state $|01101\rangle$. You can use combinadics as I suggested, or if you are working with the full space just enumerate states using binary e.g. $|0000\rangle \to 0$, $|0001\rangle \to 1$, $|0010\rangle \to 2$, $|0011\rangle \to 3$, ... $\endgroup$
    – andywiecko
    Commented Oct 17 at 11:35
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    $\begingroup$ @CuriousMind Oh, I see! So, you allow multiple particles at a given site, which makes the problem more complicated. I would suggest constructing two mappings and storing them in memory: $(i \to \text{state})$ and $(\text{state} \to i$). Assuming you have $L$ sites and $N$ particles, you can enumerate these states using combinations with replacement (or $N$-multisets) and then discard the states that don't satisfy your constraint on the maximum number of particles per site. $\endgroup$
    – andywiecko
    Commented Nov 15 at 12:10
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    $\begingroup$ For example, let's assume $L = 4$ and $N = 2$. 1. Generate all $N$-multisets for $[0, 1, ..., L - 1]$. These sets represent the sites where particles will be placed. $$ (0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3) $$ 2. Construct the states. For example, $(1, 3)$ means particles are placed at site 1 and site 3, leading to the state $|0101\rangle$. $\endgroup$
    – andywiecko
    Commented Nov 15 at 12:11
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    $\begingroup$ $$ 0 \to (0, 0): |2000\rangle, \\ 1 \to (0, 1): |1100\rangle, \\ 2 \to (0, 2): |1010\rangle, \\ 3 \to (0, 3): |1001\rangle, \\ 4 \to (1, 1): |0200\rangle, \\ 5 \to (1, 2): |0110\rangle, \\ 6 \to (1, 3): |0101\rangle, \\ 7 \to (2, 2): |0020\rangle, \\ 8 \to (2, 3): |0011\rangle, \\ 9 \to (3, 3): |0002\rangle. $$ In this example, I haven't constrained the maximum number of particles per site. In step 2, you could discard states that exceed your maximum particle constraint for each site. $\endgroup$
    – andywiecko
    Commented Nov 15 at 12:11
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    $\begingroup$ 3. Construct matrix. To calculate an element like $H_{4, 3} = \langle 0200| H |1001\rangle$, apply the appropriate operators to compute the matrix elements. $\endgroup$
    – andywiecko
    Commented Nov 15 at 12:11

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