Symmetries in the Hubbard model

Question

I would like to understand both in an intuive and in a mathematical way the meaning of the sentence "The Hubbard Hamiltonian has an SU(2) symmetry". What are the symmetry transformations that leave the Hubbard Hamiltonian unchanged?

Moreover, I would like to understand how is it possible to generalize the Hubbard model to the case of SU(N) symmetry. In particular, what does N stand for? What are the conserved quantities of the SU(2) Hubbard model and of the SU(N) Hubbard model?

user245141 · Accepted Answer

The Hubbard model is described by nearest-neighbor hopping $H_t = -tsum [c^{dagger}_{sigma,i}c_{sigma,j}+rm{h.c.} ]$, which preserves the spin-orientation, plus an on-site Coulomb replusions $H_U = U sum n_{i,uparrow}n_{i,downarrow}$. Both of these terms are invariant to a global rotation of the spins. The first term will still describe each flavor of spins hopping from site to site, and the other will still be $U$ if there are two electrons on the site and zero otherwise. Therefore, a global rotation of the spins, which is described by $SU(2)$, leaves the Hamiltonian invariant. This helps us as we can characterize the states with total spin and total $S_z$, which are conserved.

In order to expand to an $SU(N)$ symmetry, we need to characterize the particles with higher degrees of freedom, such that their symmetry is not spin-rotations of $SU(2)$ but a general $SU(N)$.

Edit after question in comments:

For $SU(N)$ it's not enough to have larger spins. An example of how to generate $SU(N)$ might be along the following lines - instead of just one type of electrons let's say we have $N$-types of electrons, each coming from a different source. For example, in the 1d Hubbard model, we can proximitize $N$ wires of spinless electrons. So now instead of labeling each by the spin index we label it by the index $nu=1,ldots,N$ that tells us at which wire the electron is. We can now define "rotations" in the basis of these wires, which simply signify basis change. These are rotations in $N$-dimensional space, that will be characterized by the symmetry group $SU(N)$. Depending on the type of Hamiltonian and which terms we allow, it might be symmetric under this rotations (that is - it will not care in which basis we write our electrons). We usually call this extra degree of freedom "flavor".

Andrew Yuan · Answer

Let me give a completely mathematical description, since I think the accepted answer is more physically motivated.
Let $Uin SU(2)$. Then its second-quantization $W_U$ on the $n$-particle Fock space would be just $Uotimescdotsotimes U$. In particular, if $U=e^{-iS}$ for some $Sin i text{su}(2)$ (its Lie algebra), then $W_U = exp(-ic^*S c)$ where $c^* S c$ is the Jordan map of $S$, i.e. short-hand for
$$
c^* Sc=sum_varphi c^*(Svarphi) c(varphi) = Sotimes Iotimes cdots otimes I+cdots+Iotimescdotsotimes Iotimes S
$$
where the summation is over any orthonormal basis $varphi$ of the single-particle Hilbert space $mathscr{H}$. It should be noted that
$$
W_U c^*(varphi)W^*_U= c^*(Uvarphi)
$$
Also notice that if $n=n_uparrow +n_downarrow$, then
$$
n^2 =n+2n_uparrow n_downarrow
$$
Also notice that $n=c^* c$ and that
$$
W_U nW_U^*=sum_sc^*(Us)c(Us)
$$
SInce $U$ is unitary, we see that $Us$ is just another orthonormal basis. Since the I already claimed (and can be easily proven) that the shorthand $c^*c$ is independent of orthonormal basis, we see that
$$
W_U n W_U^*=n
$$
Hence,
$$
W_U n_uparrow n_downarrow W_U^* = n_uparrow n_downarrow
$$
Therefore, the interaction term is $SU(2)$-invariant. You could do something  similar with the hopping term. Also, I ignored the lattice sites in my derivation, but the same proof with slight modifications hold in the general case.
From this proof, one thing becomes clear. If you want to generalize this interaction term to $SU(N)$, we would need an interaction term that looks like $n^2-n$ (or some other function of $n=c^* c$).

Symmetries in the Hubbard model

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