---
abstract: 'The bosonic $su(n)$ Hubbard model was recently introduced. The model was shown to be integrable in one dimension by exhibiting the infinite set of conserved quantities. I derive the $R$-matrix and use it to show that the conserved charges commute among themselves. This new matrix is a non-additive solution of the Yang-Baxter equation. Some properties of this matrix are derived.'
author:
- |
    [**Z. Maassarani**]{}[^1]\
    \
    [Département de Physique, Pav. A-Vachon]{}\
    [Université Laval, Ste Foy, Qc, G1K 7P4 Canada]{}[^2]\
title: 'Exact integrability of the $su(n)$ Hubbard model'
---

23.50cm -1.7cm 0.6cm

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PACS numbers: 75.10.-b, 75.10.Jm, 75.10.Lp\
Key words: Hubbard model, $su(n)$ spin-chain, integrability

October $6^{\rm th}$ 1997\
LAVAL-PHY-25/97\

Introduction
============

The two-dimensional Hubbard model was introduced to describe the effects of correlation for $d$-electrons in transition metals [@guhu]. It was then shown to be relevant to the study of high-$T_c$ superconductivity of cuprate compounds.

In one dimension the model is integrable [@liwu; @sh12; @woa]. The integrability framework of the model is the quantum inverse scattering method [@qism]. However, despite sharing many properties with discrete quantum integrable models, the model has a peculiar integrable structure which defines a class of its own.

In seeking to generalize the Hubbard model in any dimension, it was therefore natural to look for a one-dimensional generalization which is integrable. An $n$-state generalized model which contains the usual $su(2)$ model was recently introduced in [@hn]. This $su(n)$ Hubbard model was shown to possess an infinite set of conserved charges and to have an extended $su(n)$ symmetry. The model is built by coupling two copies of the recently discovered $su(n)$ XX ‘free-fermions’ model [@mm]. For $n=2$ a fermionic formulation exists, but for $n > 2$ finding an analogous framework is a tantalizing problem.

In this work I derive the $R$-matrix of the model; this provides a direct proof of the commutation of the conserved charges among themselves. Section two gives the definition of the bosonic Hamiltonian and the transfer matrix. The $R$-matrix intertwining the monodromy matrices is derived in section three. In section four some properties of this new matrix are given. I conclude with some remarks and outline some outstanding issues.

The model
=========

Let $E^{\af\be}$ be the $n\times n$ matrix with a one at row $\af$ and column $\be$ and zeros otherwise. The $su(n)$ Hubbard Hamiltonian on a ring then reads [@hn]: H\_2 &=&\_i h\_[ii+1]{} +\_i h\^[’]{}\_[ii+1]{} + U\_i h\^c\_i\[h2\]\
&=& \_i \_[&lt; n]{} (x E\_i\^[n]{} E\_[i+1]{}\^[n]{} + x\^[-1]{} E\_i\^[n]{} E\_[i+1]{}\^[n]{} + (EE\^[’]{})) + U \_i (\_i +) (\^[’]{}\_i +)where $\rho = \sum_{\af < n} E^{\af\af} -(n-1) E^{nn}$, and primed and unprimed quantities correspond to two commuting copies of the $E$ matrices. The Hamiltonians $h$ and $h^{'}$ are $su(n)$ XX Hamiltonians [@mm]. The complex free parameter $x$ is a deformation inherited from the XX model. The Hamiltonian $H_2$ is defined in one dimension but can be evidently defined on any lattice; integrability is lost however.

For $n=2$ and $x=1$, and using Pauli matrices, the Hamiltonian is just the integrable bosonic version of the usual Hubbard Hamiltonian [@sh12]: H\_2\^[(2)]{}= \_i (\^x\_i \^x\_[i+1]{} + \^y\_i \^y\_[i+1]{}) + (\^[’]{} ) + U\_i \^z\_i\^[’z]{}\_i The Hamiltonians can be written simply in terms of $su(n)$ hermitian traceless matrices. For $|x|=1$ the Hamiltonians are hermitian.

The transfer matrix is the generator of the infinite set of conserved quantities. Its construction was given in [@hn]. We recall it here. Consider first the $R$-matrix of the $su(n)$ XX model [@mm]: R() &=& a() \[E\^[nn]{}E\^[nn]{}+\_[[, &lt;n]{}]{} E\^E\^\]\
& & + b()\_[[&lt;n]{}]{}(x E\^[nn]{}E\^ + x\^[-1]{} E\^E\^[nn]{})\
& & + c() \_[[&lt;n]{}]{}(E\^[n ]{}E\^[n]{} + E\^[n]{}E\^[n]{}) where $a(\la)=\cos(\la)$, $b=\sin(\la)$ and $c(\la)=1$. The functions $a$, $b$ and $c$ satisfy the ‘free-fermion’ condition: $a^2 +b^2 = c^2$. For this set of parameters, a Jordan-Wigner transformation turns the $U=0$ Hamiltonian density for $su(2)$ into a fermionic expression for free fermions hopping on the lattice.

Consider also the matrix I\_0 (h) =() + () C\_0 C\^[’]{}\_0 =( C\_0 C\^[’]{}\_0) where $C=\sum_{\af < n} E^{\af\af}-E^{nn}$. We stress that $C$ turns out to be the fundamental matrix, not the $su(n)$ generator $\rho$. We have $\rho +\frac{n-2}{2}{\rm Id} = \frac{n}{2} C$, for $n \geq 2$. The parameter $h$ is related to the spectral parameter $\lambda$ by (2h) = (2) \[rela\] One chooses for $h(\la)$ the principal branch which vanishes for vanishing $\la$ or $U$. Then for $U=0$ the monodromy matrix becomes a tensor product of two uncoupled XX models. The Lax operator at site $i$ is given by: L\_[0i]{} () = I\_0(h) R\_[0i]{}() R\^[’]{}\_[0i]{}() I\_0(h) and the monodromy matrix is a product of Lax operators, $T(\lambda)= L_{0M}(\la)...L_{01}(\la)$, where $M$ is the number of sites on the chain. The transfer matrix is the trace of the monodromy matrix over the auxiliary space 0: $\tau (\la)= {\rm Tr}_0 \;\left[\left( L_{0M}...L_{01}\right)(\la)\right]$. One possible set of conserved quantities is given by H\_[p+1]{} = ([d\^p ()d\^p]{})\_[=0]{} The proof that $H_2$ commutes with $\tau(\lambda)$ was given in [@hn]. The derivative of the matrix $I$ gives the coupling term appearing in (\[h2\]). Note that the definition involving a logarithm has two benefits. Besides giving the most local operators, it further disentangles the two copies.

The $R$-matrix
==============

We derive the $R$-matrix intertwining two monodromy matrices at different spectral parameters. To this end we generalize the algebraic method of the Decorated Star Triangle Equation introduced by Shastry [@sh3].

The XX $R$-matrix satisfies the regularity property $\check R (0) = {\rm Id}$, the unitarity condition $\check R (\lambda) \check R (-\lambda) = {\rm Id} \;\cos^2\lambda$ and the Yang-Baxter equation \_[12]{}(-) R\_[13]{}()R\_[23]{}() = R\_[13]{}() R\_[23]{}()\_[12]{}(-)\[ybec\] where $R=P \check{R}$ and $P$ is the permutation operator on the tensor product of two $n$-dimensional spaces. It is easy to verify that it also satisfies a decorated Yang-Baxter equation \_[12]{}(+) C\_1 R\_[13]{}() R\_[23]{}() = R\_[13]{}() R\_[23]{}()C\_2\_[12]{}(+)\[dybec\]

We now look for the $R$-matrix intertwining two $L$-matrices: (\_1,\_2) L(\_1)L(\_2) = L(\_2)L(\_1) (\_1,\_2)\[rll\] The $su(2)$ case lead us to consider the following Ansatz [@sh3]: (\_1,\_2) &=& I\_[12]{}(h\_2) I\_[34]{}(h\_1) ( \_[13]{}(\_1-\_2) \_[24]{}(\_1-\_2) .\
& &.+ \_[13]{}(\_1+\_2) C\_1 \_[24]{}(\_1+\_2)C\_2 ) I\_[12]{}(-h\_1) I\_[34]{}(-h\_2) The $R$-matrix acts on the product of four auxiliary spaces labeled from 1 to 4, and $\alpha$, $\beta$ are to be determined. One then requires relation (\[rll\]) to be satisfied and uses (\[ybec\]) and (\[dybec\]) to derive the following equation: &(\_[13]{}(\_1-\_2) \_[24]{}(\_1-\_2) + C\_3\_[13]{}(\_1+\_2)C\_4 \_[24]{}(\_1+\_2) )I\_[12]{}(2h\_1) I\_[34]{}(2h\_2) =&\
&I\_[12]{}(2h\_2) I\_[34]{}(2h\_1) ( \_[13]{}(\_1-\_2) \_[24]{}(\_1-\_2) + \_[13]{}(\_1+\_2) C\_1 \_[24]{}(\_1+\_2)C\_2 )& Expanding the exponentials and taking into account all the terms yield only two equations: = (h\_1+h\_2) , = (h\_1-h\_2) where $a=\cos(\la_1-\la_2)$, $b=\sin(\la_1-\la_2)$, $A=\cos(\la_1+\la_2)$ and $B=\sin(\la_1+\la_2)$. The compatibility equation = is satisfied provided equation (\[rela\]) is satisfied for the pairs $(\la_1,h_1)$ and $(\la_2,h_2)$. One can then pull out $\alpha=\alpha(\la_1,\la_2)$ which appears as an arbitrary normalization of the $R$-matrix, to obtain: (\_1,\_2)&=& (\_1,\_2) I\_[12]{}(h\_2) I\_[34]{}(h\_1) ( \_[13]{}(\_1-\_2) \_[24]{}(\_1-\_2) +.\
& &. (h\_1+h\_2) \_[13]{}(\_1+\_2) C\_1 \_[24]{}(\_1+\_2)C\_2 ) I\_[12]{}(-h\_1) I\_[34]{}(-h\_2)\[rc\]

The monodromy matrix being a tensor product of $M$ copies of $L$ matrices, one has (\_1,\_2) T(\_1)T(\_2) = T(\_2)T(\_1) (\_1,\_2)\[rtt\] Taking the trace over the auxiliary spaces and using the cyclicity property of the trace one obtains $[\tau(\la_1),\tau(\la_2) ]=0$. We have thus proven that all the conserved charges $H_p$ mutually commute.

Note that this proof is rigorous and valid [*for all values of*]{} $n$, and for arbitrary values of the complex parameter $x$. It only involves the algebraic properties of the operators appearing in the various matrices, not the specific $n$-dependent representation. The equations (\[ybec\]) and (\[dybec\]) are the only equations of this type needed for the proof.

Properties of the $\check{R}$ matrix
====================================

I now give some properties of the $R$-matrix. At $U=0$ the two $XX$ models decouple and $h(\la,U)=0$. Expression (\[rc\]) indeed decouples as a tensor product of two $su(n)$ XX $\check{R}$-matrices.

The matrix also satisfies the regularity property (\_1,\_1) = (\_1,\_1) and the unitarity property: (\_1,\_2) (\_2,\_1)& =& \^2(\_1,\_2) \^2(\_1-\_2)\
&&(\^2(\_1-\_2) - \^2(\_1+\_2) \^2(h\_1-h\_2)) [Id]{} The derivation of the last property is straightforward and involves algebraic relation between the building blocks of the $su(n)$ XX $\check{R}$-matrix.

One can invoke the associativity of the algebra of $L$ matrices, which ultimately reduces to the associativity of usual matrix multiplication, to conclude that the intertwiner satisfies a Yang-Baxter relation of its own. The two ways of permuting a product of three $L$-matrices imply & L(\_1)L(\_2)L(\_3) \^[-1]{}= L(\_1)L(\_2) L(\_3) &\
& = ( \_[12]{}(\_2,\_3) \_[23]{}(\_1,\_3) \_[12]{}(\_1,\_2) )\^[-1]{} \_[23]{}(\_1,\_2) \_[12]{}(\_1,\_3) \_[23]{}(\_2,\_3) I am unaware of the existence of an equivalent of the Schur lemma for the algebra of $L$-matrices. This would allow to conclude that $\check{\Pi}\propto {\rm Id}$. Once proportionality is established, the regularity property ensures that the proportionality constant is one. We can argue that $\check{\Pi} =  {\rm Id}$ holds because it has been explicitly verified for $n=2$ [@sw], and because the building blocks of the matrix satisfy algebraic relations which are independent of $n$. Thus the $R$-matrix satisfies the Yang-Baxter equation: \_[12]{}(\_2,\_3) \_[23]{}(\_1,\_3) \_[12]{}(\_1,\_2) = \_[23]{}(\_1,\_2) \_[12]{}(\_1,\_3) \_[23]{}(\_2,\_3)\[rybe\] where $\la$ and $h$ are related through (\[rela\]).

Using the explicit expression (\[rc\]), it should be possible and it is instructive to try to check that the above equation is satisfied for any value of $n$. The factors $I$ drop out and half of the terms on both sides of the YBE compensate each other because relations (\[ybec\]) and (\[dybec\]) hold. The eight remaining terms involve highly non-trivial relations. Each term is a product of six $R$-matrices, three for every copy, in the ordering dictated by the YBE. The $C$ factors can be dropped by changing the arguments of the $R$-matrices appropriately. However the arguments do not allow the use of (\[ybec\], \[dybec\]) because the middle argument is not the sum of the extreme ones. One is forced to expand all the products on a basis and to regroup terms and check that the resulting trigonometric constraints are satisfied. I have not verified whether all these relations hold. Although specific particularities pertaining to the XX matrix are needed, I stress again that the proof is algebraic. In this respect the proof of [@sw] for $n=2$, although following a different approach, should generalize in a straightforward way to any value of $n$.

Conclusion
==========

We have shown that all the conserved charges of the $su(n)$ Hubbard model mutually commute by exhibiting the intertwining matrix. This matrix is the $su(n)$ generalization of the $su(2)$ one obtained in [@sh12]. Some properties where then derived. A notable feature of the matrix is its non-additivity property; the $\la$ dependence cannot be reduced to a difference $(\la_1-\la_2)$.

One can now start diagonalizing the Hamiltonians by the method of the algebraic Bethe Ansatz. Preliminary results suggest an interesting structure for the Bethe eigenstates [@win].

One can also consider the extension of the $su(n)$ model to other algebras. The algebraic underlying structure of the $su(n)$ XX model should admit generalizations [@mm; @win].

 [**Acknowledgement:**]{} I thank P. Mathieu for bringing to my attention reference [@sh3] and for continued support.

 [**Note:**]{} While this work was being written, Martins exhibited a gauge-transformed version of the foregoing $\check{R}(\la_1,\la_2)$ matrix. The derivation is also based on a generalization of Shastry’s method. However, the proof in [@mjm] was carried out for $n=3$, and ‘extensive checks’ were made for $n=3,4$. The expression of $\check{R}$ for all values of $n$ is left as a (correct) [*conjecture*]{}. This reference also used unnecessarily complicated versions of the Yang-Baxter equations.

[30]{}

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[^1]: Work supported by NSERC (Canada) and FCAR (Québec).

[^2]: email address: zmaassar@phy.ulaval.ca
