# Integrability of generalized Richardson-Hubbard model

Recently I got a bit interested in the possibility of finding spectrum of few interesting class of lattice quantum mechanical hamiltonians like Richardson's pairing hamiltonian, 1D Hubbard hamiltonian, 1D Heisenberg spin chains.

In this context I have this question, is the following generalized Richardson-Hubbard model hamiltonian exactly solvable (analytical feasibility of finding the spectrum) or integrable :

$$\mathbb{\hat{H}}=\sum_{x}^{}\hat{\mathbf{\Psi}}_{}^{\dagger}(x)\mathbb{A}(x)\hat{\mathbf{\Psi}}_{}^{}(x)+\sum_{x,y}\hat{\mathbf{\Psi}}_{}^{\dagger}(x)\otimes\hat{\mathbf{\Psi}}_{}^{\dagger}(x)\mathbb{B}(x,y)\hat{\mathbf{\Psi}}_{}^{}(y)\otimes\hat{\mathbf{\Psi}}_{}^{}(y)$$

where

$$\Psi(x)=\begin{pmatrix} c_{1}^{}(x) & \cdots & c_{n}^{}(x) & c_{1}^{\dagger}(x) & \cdots & c_{n}^{\dagger}(x)\end{pmatrix}_{}^{T}$$

with $$c_{k}^{\dagger}(x)/c_{k}^{}(x)$$ being fermionic creation/annhilation operator for creating/annhilating fermion of flavor $$k$$ at the site $$x$$ of a $$1D$$ periodic lattice for example. Further $$\mathbb{A}(x)$$ and $$\mathbb{B}(x,y)$$ are complex $$n \times n$$ and $$n_{}^{2} \times n_{}^{2}$$ matrices respectively. Just to be more general, here restriction to hermitian nature of $$\mathbb{\hat{H}}$$ is not assumed (so that it is possible to generalize techniques to open quantum systems setup).

If as such $$\mathbb{\hat{H}}$$ is not exactly solvable/integrable, what further restrictions are necessary on $$\mathbb{A}(x)$$ and $$\mathbb{B}(x,y)$$ for this problem to be exactly solvable/integrable : like restricting $$x,y$$ in the second summation of the hamiltonian definition to nearest neighbour and/or restricting $$\mathbb{A}(x)$$ and $$\mathbb{B}(x,y)$$ to be homogeneous (independent of $$x$$ and $$y$$) and/or imposing restrictions on structure of $$\mathbb{A}(x)$$ and/or $$\mathbb{B}(x,y)$$ matrices and/or dimensionality of flavor space ($$n$$) and so on (excluding trivial limits like non-interacting case ($$\mathbb{B}(x,y)=\mathbb{O}_{n_{}^{2}\times n_{}^{2}}^{}$$ for all $$x,y$$ of lattice) and/or all lattice sites decoupled case ($$\mathbb{B}(x,y)=\mathbb{B}\delta_{xy}^{}$$ - note here that i am having small $$n$$ case in mind of order atleast 4 and not exceeding 6)). One particular case I am interested in is $$\mathbb{A}(x)$$ and $$\mathbb{B}(x,y)$$ are homogeneous and $$n=4$$.

Specifically i am looking for the amenability of Richardson's ansatz or coordinate/algebraic/functional Bethe ansatz methods (i am still trying to figure out basic elements of algebraic Bethe ansatz, references along this direction will be extremely helpful too) to the exact solvability/integrability of $$\mathbb{\hat{H}}$$.

• There are a few options for making integrable Hamiltonians with algebraic Bethe ansatz. You could try fusion procedure to see that neighbor condition is not necessary. Play with using different representations of the requisite quantum group on the different sites and you see you can avoid homogeneity. – AHusain Sep 24 '18 at 0:27
• @AHusain Thank you for the helpful comment. I am a bit novice to Bethe ansatz methods, can you suggest any helpful reference along the direction of your comment? – Sunyam Sep 24 '18 at 0:31
• Lectures on Integrability of 6-Vertex Model – AHusain Sep 24 '18 at 3:05

However, the model that you wrote down looks like multi-component Yang-Gaudin Fermi gases, if $$\mathbb{A}(x)$$ and $$\mathbb{B}(x,y)$$ are chosen that all different species of fermions have the same mass, and the interaction among them are delta interaction with the same strength. The most physically relevant model here is two-component Yang-Gaudin model, solved via coordinate Bethe ansatz by CN Yang and Michel Gaudin almost simultaneously. There is a recent review article on that:arxiv: 1310.6446.
• I intended to mention (but i missed to state it explicitly) in the question, matrix $\mathbb{A}(x)$ is supposed to have scalar (opposed to differential differential operators as entries) and further for this case when $\mathbb{B}(x,y)$ is assumed to be contact like (i.e., $\mathbb{B}(x,y)=\mathbb{B}\delta(x,y)$ which i was excluding), at least for small $n$ of order $4$, the problem is straight forward as different sites are decoupled. What i am actually interested is the case $\mathbb{A}(x)=\mathbb{A}$, $\mathbb{B}(x,y)=\mathbb{B}$ with $n=4$. Any comments on this case? – Sunyam Oct 23 '18 at 16:53
• I don't quite understand your setup. Following your example, the Hamiltonian is completely decoupled from site to site, as you mentioned. Therefore, for any $n$, one just have to calculate spectrum from each site (the chemical potential and Hubbard interaction commute) and add them up . There is no need to discuss about integrability at all. If you want that, it is exactly solvable without using any technique like Bethe ansatz. – Exhaustive Oct 25 '18 at 15:33
• That was precisely the case I was excluding it. As you said, atleast for small $n$ of order 4, the problem is trivial, for which integrability discussion is a bit an overkill. What I am actually interested in is the case stated in the comment above (the case where all the sites are interacting with each other through generalized two body potential, with generalized one body part of the hamiltonian being site local). – Sunyam Oct 25 '18 at 19:39