Exchange interaction

Mutual repulsion

The two-site Hubbard model demonstrates the emergence of a new low-energy scale if the mutual Coulomb repulsion $U$ of electrons is dominant. We again use the single-band Hubbard model as a basis whose Hamiltonian is

\[\begin{align} H&=U\sum_i n_{i\uparrow}n_{i\downarrow} -\sum_{\langle ij\rangle,\sigma}t_{ij}c_{i\sigma}^\dagger c_{j\sigma}, \quad t_{ii}=0 \\ &=H_0+H_1, \\ H_1&=T_{-1}+T_0+T_1, \\ [H_0,T_m]&=mUT_m. \end{align}\]

For $t_{ij}=0$ the Hamiltonian commutes with the particle number operator, therefore we can classify the eigenstates of it according to the number of doubly occupied sites. The energy difference between a state with $M+1$ doubly occupied sites and a state with $M$ doubly occupied sites is just $U$. Therefore the ground state will be a linear combination of wavefunctions with the minimal number of double occupancies. At half filling for example each of the N lattice sites is occupied by exactly one electron and $M=0$. Because each electron carries a spin with two possible orientations, this state is $2^N$ times degenerate.

Finite kinetic energy

For $t_{ij}\ne0$, $M$ is no longer a good quantum number. However, creating a double occupancy still costs $U$, thus for small $t_{ij}$ the projection onto a subspace with constant $M$ will continue to be a good approximation. For a particle number less than or equal to $N$, we can assume $M=0$ contains the ground state and low-lying excitations.

So how can we boil down or Hamiltonian to its low-energy representation containing only these $M=0$ states? With the help of a unitary transformation

\[\begin{align} H_\text{eff}&=S^\dagger HS,\quad S={\rm e}^A,\quad A^\dagger=-A \\ &= H +\left[A,H\right] +\frac{1}{2}\left[A,\left[A,H\right]\right] +\ldots \end{align}\]

we construct the antiunitary operator

\[A=\sum_{i=1}^\infty\lambda^iA_i\]

choosing $A_i$ such that those terms changing $M$ can be eliminated recursively order by order in the above expansion [1,2,3] with

\[\begin{align} S^\dagger HS &= H_0+\lambda\left(H_1+\left[H_0,A_1\right]\right) \\ &+ \lambda^2\left(\left[H_0,A_2\right]+\left[H_1,A_1\right] +\frac12\left[\left[H_0,A_1\right],A_1\right]\right) \\ &+ \lambda^3\left\{\vphantom{\frac12} \left[H_0,A_3\right]+\left[H_1,A_2\right]\right. \\ &+\frac12\left( \left[\left[H_0,A_1\right],A_2\right] +\left[\left[H_0,A_2\right],A_1\right] +\left[\left[H_1,A_1\right],A_1\right] \right) \\ &+\left. \frac16\left[\left[\left[H_0,A_1\right],A_1\right],A_1\right] \right\} \\ &+\dots \end{align}\]

Choosing

\[\begin{align} A_1&=\frac1U\left(T_{-1}-T_1\right), \\ A_2&=\frac1{U^2}\left[T_0,T_{-1}+T_1\right], \end{align}\]

we obtain in the $M=0$ subspace

\[\begin{align} S^\dagger HS &= H_0+\lambda T_0 -\frac{\lambda^2}U\left[T_{-1},T_1\right] \\ &+ \frac{\lambda^3}{2U^2}\left( \left[T_{-1},\left[T_0,T_1\right]\right] +\left[T_{1},\left[T_0,T_{-1}\right]\right] \right) \\ &+{\cal O}\left(\lambda^4\right). \end{align}\]

Expressed with the fermion operators, to lowest order in $1/U$ we obtain

\[\begin{align} H_\text{eff}&= -\sum_{\langle ij\rangle\sigma}t_{ij} {\cal P}c_{i\sigma}^\dagger c_{j\sigma}{\cal P} \\ &+\frac4U\sum_{\langle ij\rangle}\left|t_{ij}\right|^2 \left({\bf S}_{i}{\bf S}_{j} -\frac14n_{i}n_{j}\right) \\ &-\frac1U\sum_{\langle ijk\rangle\sigma} t_{ik}t_{jk}^* \left( {\cal P}c_{j\sigma}^\dagger n_{k\bar{\sigma}}c_{i\sigma}{\cal P} +{\cal P}c_{j\sigma}^\dagger c_{k\bar{\sigma}}^\dagger c_{k\sigma}c_{i\bar{\sigma}}{\cal P} \right) \end{align}\]

where $\cal P$ is a projector onto states without doubly occupied sites and

\[{\bf S}_{i}=\frac12\sum_{\alpha\beta} c_{i\alpha}^\dagger{\bf \sigma}_{\alpha\beta}c_{i\beta}\]

is the usual spin operator on site $i$. The first term in the expansion above resembles the kinetic energy and reflects the fact that an electron can only hop onto a neighboring site if this is empty, commonly called correlated hopping. The second term is comprised of a spin exchange part and a counting term with an attractive nearest-neighbor interaction. The last term of the effective Hamiltonian describes hopping processes involving three adjacent sites: An electron hops from site $i$ to site $k$, then however not back to site $i$ (that’s the spin term describing this). Rather an electron hops from site $k$ to site $j$ different from $i$, either having the same spin orientation as the "original" one or accompagnied by a spin flip.

Exchange interaction

At half filling, we have $n_i=1$, $M=0$ at each site, hopping is impossible, thus

\[H_{\text{eff}}=\sum_{\langle ij\rangle}J_{ij} {\bf S}_{i}{\bf S}_{j}+\text{const},\quad J_{ij}=\frac{4|t_{ij}|^2}U\]

which is, apart from a constant, the famous Heisenberg Hamiltonian with exchange energy $J_{ij}=4|t_{ij}|^2/U$.

To summarize, physically the impact of a finite kinetic energy of the electrons characterized by $t_{ij}$ in the presence of large Coulomb repulsion $U$ is thus to create spin excitations with characteristic energy $J_{ij}$ opening the door to a wide variety of truly quantum many-body phenomena, be it magnetic order, heavy fermions, or new kinds of disordered ground states. We shall discuss examples for these elsewhere on this site.

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[1] A. B. Harris and R. V. Lange: Single-Particle Excitations in Narrow Energy Bands. Phys. Rev. 157, 295 (1967).

[2] A. H. MacDonald, S. M. Girvin, and D. Yoshioka: t/U expansion for the Hubbard model. Phys. Rev. B 37, 9752 (1988).

[2] B. Schmidt: Theorie des Halbmetalls Yb₄As₃. Thesis, Stuttgart 1996 and references therein.