Reading Notes on Canonical Transformation of Fermions

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Summary of the paper (Ostlund and Mele, PRB 1991).

This paper proved that non-linear canonical transformations of fermionic basic operators at one local point form a Lie group of $SU(2) \otimes SU(2) \otimes U(1) \otimes \mathbb{Z}_2$. The authors tried to enumerate all possible point-local gauge transformation acting on spin-1/2 operators. They applied their findings to identifying subgroups that form local and global gauge symmetries of the Hubbard-Heisenberg model.

Comment: This paper only tried to use the symmetries to analyze a Hamiltonian. I am more interested in whether we can find a better representation to reduce the computational complexity.

For simplicity, we use basic operators to denote creation and annihilation operators.

Formulations of the Local Canonical Transformation

A natural basis of a $4\times 4$ matrix can be composed of 16 basis matrices: ${m(i,j)}_{i,j=1}^4$, where $m(i,j)$ is a matrix satisfying $[m(i,j)]_{ij} = 1$, and all other elements are equal to 0. A $SU(4)$ matrix $x$ is a $4\times 4$ matrix that satisfies $x^\dagger x = x x^\dagger = I$ and $\det(x) = 1$.

We choose the Fock vectors of a single-site system as the basis: $\psi_1 = |0\rangle$, $\psi_2 = |\uparrow\downarrow\rangle$, $\psi_3 = |\uparrow\rangle$, $\psi_4 = |\downarrow\rangle$. Then $m(i,j)$ can be represented by the non-linear operations onto the Fock basis:

\[m(i,j) \psi_k = \psi_i \delta_{jk}\]

And the super-matrix of $m(i,j)$ can be represented by a matrix of non-linear operators

\[m = \begin{pmatrix}1 - n_\uparrow - n_\downarrow + n_{\uparrow}n_{\downarrow} & c_{\downarrow}c_{\uparrow} & (1 - n_\downarrow)c_{\uparrow} & (1 - n_\uparrow)c_{\downarrow} \\ c_{\uparrow}^\dagger c_{\downarrow}^\dagger & n_{\uparrow}n_{\downarrow} & -c_{\downarrow}^\dagger n_{\uparrow} & c_{\uparrow}^\dagger n_\downarrow\\ c_{\uparrow}^\dagger (1 - n_\downarrow) & -n_{\uparrow}c_{\downarrow} & n_\uparrow(1 - n_\downarrow) & c_{\uparrow}^\dagger c_{\downarrow} \\ c_{\downarrow}^\dagger(1 - n_\uparrow) & n_\downarrow c_{\uparrow} & c_{\downarrow}^\dagger c_{\uparrow} & n_\downarrow (1 - n_\uparrow)\end{pmatrix}\]

where $c^\dagger_\sigma$ and $c_\sigma$ are the basic operators on this site. Note that each element of $m$ is a $4\times 4$ matrix, so it is effectively a $16\times 16$ matrix.

With the above settings, we can define the general form of a canonical transformation for creation and annihilation operators of a single-site system as

\[X = \sum_{ij} x_{ij} m(j,i) = \text{Tr}(xm) \equiv P(x)\]

where $x$ is an SU(4) matrix, so $x_{ij}$ is a scalar, while $m(j,i)$ is a $4\times 4$ matrix, or a one-site operator.

Properties of $m$ and $X$

  1. $[m(i,j)]^\dagger = m(j, i)$
  2. $m(i,j)m(k,l) = \delta_{jk}m(i,l)$
  3. $\sum_{i}m(i,i) = 1$
  4. $P(x)P(y) = P(yx)$ (Comment: in the paper the r.h.s. is $P(xy)$.)
  5. $[P(x)]^\dagger = P(x^\dagger)$

Canonical transformation

The Lie bracket here is the anticommutation relation: ${c_\sigma^\dagger, c_{\sigma’}} = \delta_{\sigma,\sigma’}$, where $\sigma$ is the spin degree of freedom. Let $x$ be an SU(4) matrix, and $U = P(x)$, then the canonical transformation is

\[c_\sigma \rightarrow U^\dagger c_\sigma U, \quad c_\sigma^\dagger \rightarrow U^\dagger c_\sigma^\dagger U\]

which preserves the Lie bracket.

Comment: In the paper, it says “since no subgroup of this SU(4) commutes with all of ${c_\uparrow^\dagger, c_\downarrow^\dagger, c_\uparrow, c_\downarrow}$, there is a one-to-one relation between elements of SU(4) and the point-localized canonical transformations.”

Multi-sites

The anti-commutation relation for a multi-site system is

\[\{c_{\sigma}^\dagger(r), c_{\sigma'}^\dagger(r')\} = \delta_{\sigma, \sigma'}\delta_{r,r'}\]

The above constraint further refines $U = P(x)$ that only terms that are products of an odd number of basic operators at site $r$ are permitted. If we look at the matrix of $m$, we observe that only the two off-diagonal $2\times 2$ blocks contain products of odd number of basic operators. Therefore, the SU(4) matrix $x$ should have the form

\[x = \begin{pmatrix} ue^{i\theta/2} & -v^* e^{i\theta/2} & 0 & 0 \\ ve^{i\theta/2} & u^*e^{i\theta/2} & 0 & 0 \\ 0 & 0 & ge^{i\theta/2} & -h^*e^{i\theta/2} \\ 0 & 0 & he^{i\theta/2} & g^* e^{i\theta/2}\end{pmatrix}\]

where $|u|^2 + |v|^2 = 1$ and $|g|^2 + |h|^2 = 1$. Therefore, $x$ can be further reduced to $SU(2) \otimes SU(2) \otimes U(1)$, where $U(1)$ correspond to the phase $e^{i\theta/2}$. (Comment: there should be a typo in the paper.)

Particle-hole exchange

The entire symmetry group also contains the discrete transformations generating particle-hole exchanges separately in each spin component. However, we only need the transformations of one spin, and the other one can be derived from it.

The particle-hole exchange for the down-spin corresponds to

\[x_{\text{p-h},\downarrow} = \begin{pmatrix}0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ \end{pmatrix}\]

which leads to $P(x_{\text{p-h},\downarrow}) = c_\downarrow^\dagger + c_\downarrow$, and $P(x_{\text{p-h},\downarrow}) c_{\downarrow}^\dagger |0\rangle = |0\rangle$ (particle to hole), $P(x_{\text{p-h},\downarrow}) |0\rangle = c_{\downarrow}^\dagger|0\rangle$ (hole to particle).

The particle-hole exchange for the up-spin then can be derived by multiplying $x_{\text{p-h},\downarrow}$ by the $SU(2) \otimes SU(2) \otimes U(1)$ matrix with $u = 0, v = 1, g = 0, h = 1, \theta = 0$.

Note that $(x_{\text{p-h},\downarrow})^2 = I$, so it belongs to the $\mathbb{Z}_2$ group, so the full group of permissible unitary transformations for spin-1/2 fermions is

\[G \equiv SU(2) \otimes SU(2) \otimes U(1) \otimes \mathbb{Z}_2\]

Up to now, we have reached the first goal: to prove that non-linear canonical transformations of fermionic basic operators at one local point form a Lie group of $SU(2) \otimes SU(2) \otimes U(1) \otimes \mathbb{Z}_2$.

Natural subgroups of $G$

Next, the authors listed some natural subgroups of $G$. If not included in the notation, then the default values are $u = 1, v = 0, g = 1, h = 0, \theta = 0$, i.e., an identity matrix.

  • Pseudospin SU(2) $\equiv SU(2)_P$
\[c^\dagger_{r\uparrow} \rightarrow u c^\dagger_{r\uparrow} - v c_{r\downarrow} \\ c^\dagger_{r\downarrow} \rightarrow u c^\dagger_{r\downarrow} + v c_{r\uparrow}\]
  • Spin $SU(2) \equiv SU(2)_S$
\[c_{r,\uparrow}^\dagger \rightarrow g^* c_{r,\uparrow}^\dagger + h^* c^\dagger_{r,\downarrow}, \\ c_{r,\downarrow}^\dagger \rightarrow g c_{r,\downarrow}^\dagger - h c^\dagger_{r,\uparrow}.\]

This is the usual spin rotations.

  • Non-linear $U(1): U(1)_{NL}$
\[c_{r,\uparrow}^\dagger \rightarrow c_{r,\uparrow}^\dagger (1 - n_{\downarrow}) + e^{2i\theta} c_{r,\uparrow}^\dagger n_{\downarrow}\\ c_{r,\downarrow}^\dagger \rightarrow c_{r,\downarrow}^\dagger(1 - n_{\uparrow}) + e^{2i\theta} c_{r,\downarrow}^\dagger n_{\uparrow}\]

If there is a spin $\alpha$ on the site, then create a spin $\beta$ and gain a phase $e^{2i\theta}$; otherwise, create a spin $\beta$ with no additional phase.

  • “hole-particle up”:
\[c_{r,\uparrow}^\dagger \rightarrow c_{r, \uparrow},\\ c_{r,\downarrow}^\dagger \rightarrow c_{r, \downarrow}.\]

The spin-up particle becomes a hole.

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