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Dihedral

There are two main cases where dihedral group symmetry is relevant for physics.

Case 1: generalizing {$U(1)$} symmetry to include spin reflection. This is relevant for spin models that are anisotropic but retain spin reflection symmetry, such as the XXZ model with no external field. It could also be used for particle models at half-filling with particle-hole symmetry.

Case 2: point-group symmetry for (anti) periodic boundary conditions. This is useful for ladder models, to use a momentum space representation for each rung.

Case 1 corresponds to the group {$D_\infty$}. We look at this case first.

Group elements

Rotation of angle {$\theta$} about the z-axis, {%R(\theta) = \exp[i\theta S^z]%}

Rotation of angle {$\pi$} about the x-axis, {%T \exp[i\pi S^x]%}

These satisfy

{%T^2 = 1%} {%TR(\theta)T^{-1} = R(-\theta)%}

Representations

Following the notation of [1]. There are two 1-dimensional irred reps, {$A_{0,\nu$} where {$\nu = \pm1$}. {$A_{0,+}$} is the identity. The other reps are 2-dimensional, {$E_\mu$} where {$\mu = 1/2, 1, 3/2, \ldots$} is the 'spin' label. Reps with {$\mu$} half-integer are double-valued.

Rotation matrix elements

Start from the {$U(1)$} basis {$| m \rangle$}.

{%R(\theta) | m \rangle = \exp[i m \theta] | -m \rangle%} {%T | m \rangle = (-1)^{\xi} | -m \rangle%}

where {$\xi$} is 1/2 for a double-valued rep ({$m$} half-integer) and 0 for a single-valued rep ({$m$} integer). This gives $T^2=-1$ for a double-valued rep, as we expect for a {$2\pi$} rotation for a half-integer spin.

For {$m=0$} we need to distinguish the two representations,

{%T |0^+\rangle = |0^+\rangle%} {%T |0^-\rangle = -|0^-\rangle%} with {%R(\theta) |0\nu\rangle = |0\nu\rangle%}

Branching rules

We eventually want to develop the coupling coefficients for the group chain {$SU(2) \supset D_\infty$}. As a first step we look at how a representation {$J$} of {$SU(2)$} breaks down into reps of {$D_\infty$};

{$J$} even: {% J \rightarrow 0^+ + 1 + 2 + \cdots + J%}

{$J$} odd: {% J \rightarrow 0^- + 1 + 2 + \cdots + J%}

{$J$} half-integer: {% J \rightarrow 1/2 + 3/2 + \cdots + J%}

Irreducible matrices

From the rotation matrix elements we can now see that we must combine states {$|-m\rangle$} and {$|m\rangle$} into a single multiplet, {% ||m|\rangle = \left( \begin{array}{c} |-m\rangle \\ |m\rangle \end{array} \right)%} with rotation matrix elements {%D^{E_m}(R(\theta)) = \left(\begin{array}{cc}e^{-im\theta} & 0 \\ 0 & e^{im\theta} \end{array}\right)%} {%D^{E_m}(T) = (-1)^\xi \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)%}

Clebsch-Gordan Series

This is straightforward to construct.

{%A_{0,\nu} \times A_{0,\lambda} = A_{0,\nu\lambda}%}

{%A_{0,\nu} \times E_\mu = E_\mu%}

{%E_\mu \times E_\mu = E_{2\mu} + A_{0+} + A_{0-}%}

{%E_{\mu_1} \times E_{\mu_2} = E_{\mu_1+\mu_2} + E_{|\mu_1 - \mu_2|}%}

Clebsch-Gordan coefficients

We use notation for the projections {$m$}, for {$m$} a non-zero half-integer, and {$0+$} and {$0-$}. The corresponding representation labels are {$E_\mu$} with {$\mu = |m|$} and ${A_{0\pm}$}. Hence the representation label is uniquely determined by the projection, so we don't need to specify the rep label separately in the Clebsh-Gordan coefficient. So the notation {%C_{m_1, m_2, m}%} is sufficient, for the Clebsch-Gordan coefficient of the product of {$m_1$} and {$m_2$}. Here we assume that {$m$} is any non-zero half-integer (positive or negative), and {$\mu$} is a postive half-integer.

From [1], we state the results.

{%C_{0\nu, \mu_2, \mu} = \delta_{\mu_2,\mu}%} {%C_{0\nu, -\mu_2, -\mu} = \delta_{\mu_2,\mu} \nu%}

{%C_{\mu_1, -\mu_2, 0\nu} = \delta_{\mu_1,\mu_2}/\sqrt{2}%} {%C_{-\mu_1, \mu_2, 0\nu} = \delta_{\mu_1, \mu_2} \nu (-1)^{2 \xi} / \sqrt{2}%}

{%C_{\mu_1, \mu_2, \mu_3} = \delta_{\mu_3, \mu_1+\mu_2}%} {%C_{-\mu_1, -\mu_2, -\mu_3} = \delta_{\mu_3, \mu_1+\mu_2} (-1)^{\xi_1+\xi_2-\xi_3}%}

{%C_{-\mu_1, \mu_2, \mu_3} = \delta_{\mu_3, -\mu_1+\mu_2}%} {%C_{\mu_1, -\mu_2, -\mu_3} = \delta_{\mu_3, -\mu_1+\mu_2} (-1)^{\xi_1+\xi_2-\xi_3}%}

{%C_{\mu_1, -\mu_2, \mu_3} = \delta_{\mu_3, \mu_1-\mu_2}%} {%C_{-\mu_1, \mu_2, -\mu_3} = \delta_{\mu_3, \mu_1-\mu_2} (-1)^{\xi_1+\xi_2-\xi_3}%}

The effect of the {$(-1)^{\xi_1+\xi_2-\xi_3}$} phase factor is to give a minus sign if we are taking the product of two double-valued reps. For a double and a single, or two single-valued reps this phase factor is +1.

References

[1] Algebraic solutions for all dihedral groups, Jin-Quan Chen, Peng-Dong Fan, Luke McAven and Philip Butler, http://dx.doi.org/10.1063/1.1286513