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Assume we start with a generic many-body Hamiltonian: $$ H=\sum_{ij} t_{ij} a_i^\dagger a_j+\sum_{mnlk}U_{mnkl}a_{m}^{\dagger}a_{n}^{\dagger}a_la_k. $$ Now if there is only the one-body part, which has a matrix form we can perform a series of canonical transformations to diagonalize it using a naive but effective way like Jacob's method.

My question is for the two-body term, the operator is kind of like a (rank-4)tensor, so I am wondering whether there is a general mathematical way to diagonalize the problem like Jacob's method. After that, the Hamtonlian will have the following form: $$ H'= \sum_{I} \epsilon_I a_I^\dagger a_I+\sum_{IJ}\epsilon_{IJ}a_{I}^{\dagger}a_{J}^{\dagger}a_Ja_I+\text{higher body term} $$

ZhiYu Fan
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1 Answers1

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In general case this Hamiltonian is not exactly solvable - in fact, there are many methods and approximations developed for dealing with such Hamiltonians in various situations, just to give a few examples:

  • various mean-field approaches - like Hartree-Fock or Habbard-Stratonovich
  • perturbation methods - e.g., Dyson expansion
  • non-perturbative techniques, like renormalization group

In fact, finding the ground state of this Hamilronian is already an achievement - see Why is the ground state important in condensed matter physics?

However, in some special cases (i.e., for particular choices of $U_{mnkl}$) it can be solved exactly, using bosonization or Bethe-Ansatz.

Related
I suggest looking for inspiration (for more precise questions) in the following threads (I didn't put them in any particular order):
Is there a way to guess/find unitary transformations for Hamiltonians? Why is Hartree-Fock considered a mean-field approach?
Is Hartree-Fock a standard mean field approximation?
Higher-order perturbation in Kondo problem
What Are Some Examples of Physics Problems With Many Different Approaches That Give the Same Answer?
In what ways does quantum field theory (QFT) extend quantum mechanics (QM)? Can Fermi liquid be obtained by a canonical transformation?

Update (in response to the OP edit)
The OP suggests looking more specifically at Hamiltonian $$ H'= \sum_{I} \epsilon_I a_I^\dagger a_I+\sum_{IJ}\epsilon_{IJ}a_{I}^{\dagger}a_{J}^{\dagger}a_Ja_I+\text{higher body term} $$ Note that $$\sum_{IJ}\epsilon_{IJ}a_{I}^{\dagger}a_{J}^{\dagger}a_Ja_I= \sum_{IJ}\epsilon_{IJ}n_{I}n_{J},$$ where $n_I=a_{I}^{\dagger}a_I$ is the occupation number. The Hamiltonian without "higher body terms" (which are really still two-body, but representing the exchange interaction) is significantly simpler, and enters a number of well-known models, depending on $\epsilon_I, \epsilon_{IJ}$ and the meaning of indices $I,J$. Notably: Hubbard model, Anderson model, various models of Coulomb blockade, Tomonaga-Luttinger model, etc. Some of these are exactly solvable (Tomonaga-Luttinger, Anderson), for others there exist well developed approximate methods and non-perturbative results.

Roger V.
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