Quantum Distance
Quantum geometry tensors are derived by computing the (gauge-invariant) inner product of two quantum states transformed over a parameter space.
Quantum distance
The quantum distance is defined as the infinitesimal difference between quantum states by the variation $d\boldsymbol{\lambda}$:
\[\begin{align*} ds^{2} &= ||\ket{\psi(\boldsymbol{\lambda}+d\boldsymbol{\lambda})}-\ket{\psi(\boldsymbol{\lambda})}||^{2} = \left(\bra{\psi(\boldsymbol{\lambda})}+\frac{\partial}{\partial\boldsymbol{\lambda}}\bra{\psi(\boldsymbol{\lambda})}d\boldsymbol{\lambda}-\bra{\psi(\boldsymbol{\lambda})}\right)\left(\ket{\psi(\boldsymbol{\lambda})}+\frac{\partial}{\partial\boldsymbol{\lambda}}\ket{\psi(\boldsymbol{\lambda})}d\boldsymbol{\lambda}-\ket{\psi(\boldsymbol{\lambda})}\right) \\ &= \Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})} d\lambda^{\mu} d\lambda^{\nu} \\ &= (\gamma_{\mu\nu}+i\sigma_{\mu\nu}) d\lambda^{\mu} d\lambda^{\nu} \end{align*}\]where the real and imaginary parts of $\Braket{\partial_{\mu}\psi(\boldsymbol{\lambda})\vert\partial_{\nu}\psi(\boldsymbol{\lambda})}$ is given by $\gamma_{\mu\nu}$ and $\sigma_{\mu\nu}$, respectively:
\[\begin{align*} \gamma_{\mu\nu} &= \mathrm{Re}\left[\Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})}\right],\\ \sigma_{\mu\nu} &= \mathrm{Im}\left[\Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})}\right] \end{align*}\]Note that the $\gamma_{\mu\nu}$ is symmetric ($\gamma_{\mu\nu} = \gamma_{\nu\mu}$) and $\sigma_{\mu\nu}$ ($\sigma_{\mu\nu} = -\sigma_{\nu\mu}$) is antisymmetric since the inner product of two states is Hermitian. In addition, $\sigma_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}$ vanishes due to the antisymmetry of $\sigma_{\mu\nu}$ and the symmetry of $d\lambda^{\mu}d\lambda^{\nu}$. Thus, the distance is rewritten as:
\[\begin{equation} ds^{2} = \gamma_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu} \end{equation}\]Comment. The imaginary part; $\sigma_{\mu\nu}$ does not explicitly appear in the quantum distance. This is because quantum distance is mathematically defined in the Riemannian manifold, in which a metric must be symmetric.
Gauge invariance of $\gamma_{\mu\nu}$
In order for $ds^{2}$ to be a quantity that measures the distance between states, $ds^{2}$ must be gauge invariant. The gauge covariant property of $\gamma_{\mu}$ can be shown by applying the $U(1)$ gauge transformation to the state; $\ket{\psi’(\boldsymbol{\lambda})} = e^{i\phi(\boldsymbol{\lambda})}\ket{\psi(\boldsymbol{\lambda})}$. Firstly, the gradient of this transformed state gives:
\[\begin{equation} \Ket{\frac{\partial}{\partial\lambda_{\mu}}\psi'(\boldsymbol{\lambda})} = i\frac{\partial\phi}{\partial\lambda_{\mu}}e^{i\phi(\mathbf{\boldsymbol{\lambda}})}\ket{\psi(\boldsymbol{\lambda})}+e^{i\phi(\mathbf{\boldsymbol{\lambda}})}\Ket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})} \end{equation}\]The inner product of two transformed states is given by:
\[\begin{align*} \gamma'_{\mu\nu} &= \mathrm{Re}\left[\Braket{\frac{\partial\psi'(\boldsymbol{\lambda})}{\partial\lambda_{\mu}}|\frac{\partial\psi'(\boldsymbol{\lambda})}{\partial\lambda_{\nu}}}\right] \\ &= \left(-i\frac{\partial\phi}{\partial\lambda_{\mu}}e^{-i\phi(\mathbf{\boldsymbol{\lambda}})}\bra{\psi(\boldsymbol{\lambda})}+e^{-i\phi(\mathbf{\boldsymbol{\lambda}})}\Bra{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})}\right) \left(i\frac{\partial\phi}{\partial\lambda_{\nu}}e^{i\phi(\mathbf{\boldsymbol{\lambda}})}\ket{\psi(\boldsymbol{\lambda})}+e^{i\phi(\mathbf{\boldsymbol{\lambda}})}\Ket{\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})}\right) \\ &= \frac{\partial\phi}{\partial\lambda_{\mu}}\frac{\partial\phi}{\partial\lambda_{\nu}}-i\frac{\partial\phi}{\partial\lambda_{\mu}}\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})}+i\frac{\partial\phi}{\partial\lambda_{\nu}}\Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\psi(\boldsymbol{\lambda})}+\Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})} \\ &= \frac{\partial\phi}{\partial\lambda_{\mu}}\frac{\partial\phi}{\partial\lambda_{\nu}}-\frac{\partial\phi}{\partial\lambda_{\mu}}A_{\nu}-\frac{\partial\phi}{\partial\lambda_{\nu}}A_{\mu}+\gamma_{\mu\nu} \end{align*}\]where $\mathrm{Re}$ is omitted for simplicity and $A_{\mu}$ is the Berry connection:
\[\begin{equation} A_{\mu}(\boldsymbol{\lambda}) = i\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\mu}}|\psi(\boldsymbol{\lambda})} \end{equation}\]The gauge transformation also changes the Berry connection as:
\[\begin{align*} A'_{\mu}(\boldsymbol{\lambda}) &= i(\bra{\psi(\boldsymbol{\lambda})}e^{-i\phi(\boldsymbol{\lambda})})\left(i\frac{\partial\phi}{\partial\lambda_{\mu}}e^{i\phi(\mathbf{\boldsymbol{\lambda}})}\ket{\psi(\boldsymbol{\lambda})}+e^{i\phi(\mathbf{\boldsymbol{\lambda}})}\Ket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})}\right) \\ &= -\frac{\partial\phi}{\partial\lambda_{\mu}}+i\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})} \\ &= A_{\mu}(\boldsymbol{\lambda})-\frac{\partial\phi}{\partial\lambda_{\mu}} \end{align*}\]To make $\gamma_{\mu\nu}$ gauge-invariant; one introduces quantum metric $g_{\mu\nu}$:
\[\begin{equation} g_{\mu\nu} \equiv \gamma_{\mu\nu}-A_{\mu}(\boldsymbol{\lambda})A_{\nu}(\boldsymbol{\lambda}) \end{equation}\]where the second term cancels out the terms in $\gamma’_{\mu\nu}$, except $\gamma_{\mu\nu}$. This is because the product of two Berry connections is transformed as:
\[\begin{align*} A'_{\mu}(\boldsymbol{\lambda})A'_{\nu}(\boldsymbol{\lambda}) &= \left(A_{\mu}(\boldsymbol{\lambda})-\frac{\partial\phi}{\partial\lambda_{\mu}}\right)\left(A_{\nu}(\boldsymbol{\lambda})-\frac{\partial\phi}{\partial\lambda_{\nu}}\right) \\ &= A_{\mu}(\boldsymbol{\lambda})A_{\nu}(\boldsymbol{\lambda})-A_{\mu}(\boldsymbol{\lambda})\frac{\partial\phi}{\partial\lambda_{\nu}}-A_{\nu}(\boldsymbol{\lambda})\frac{\partial\phi}{\partial\lambda_{\mu}}+\frac{\partial\phi}{\partial\lambda_{\mu}}\frac{\partial\phi}{\partial\lambda_{\nu}} \end{align*}\]The difference between $\gamma_{\mu\nu}$ and $g_{\mu\nu}$ lies in the vector space of quantum states: $\gamma_{\mu\nu}$ measures the distance of bare states in Hilbert space $\mathcal{H}$, while $g_{\mu\nu}$ measures the distance of rays in Hilbert space $\mathcal{H}/U(1)$. In addition, due to the principle of gauge invariance in physics, observables that are Hermitian operators act on rays. Thus, the main interest is in $g_{\mu\nu}$, not $\gamma_{\mu\nu}$.
Quantum geometric tensor
In a more general context, the gauge invariant form of $\Braket{\partial_{\mu}\psi(\boldsymbol{\lambda})\vert\partial_{\nu}\psi(\boldsymbol{\lambda})}$ is referred to as a quantum geometric tensor (QGT), which is given by:
\[\begin{equation} \mathfrak{G}_{\mu\nu}(\boldsymbol{\lambda}) \equiv \Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})} - \Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\psi(\boldsymbol{\lambda})}\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})} \end{equation}\]It can be separated into its real part $g_{\mu\nu}$ and imaginary part $\Omega_{\mu\nu}$:
\[\begin{equation} g_{\mu\nu} = \mathrm{Re}[\mathfrak{G}_{\mu\nu}],~\mathrm{and}~\Omega_{\mu\nu} = \mathrm{Im}[\mathfrak{G}_{\mu\nu}] \end{equation}\]Relationship between $g_{\mu\nu}$ and quantum distance
Recall that the quantum distance is determined by the real part of $\Braket{\partial_{\mu}\psi(\boldsymbol{\lambda})\vert\partial_{\nu}\psi(\boldsymbol{\lambda})}$, which had to be transformed into the real part of $\mathfrak{G}_{\mu\nu}$ to be gauge invariant. This statement is rewritten as:
\[\begin{equation} ds^{2} = g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu} \end{equation}\]To explicitly see how $g_{\mu\nu}$ plays the role of a metric, one takes the inner product of a state $\ket{\psi(\boldsymbol{\lambda})}$ and with $\ket{\psi(\boldsymbol{\lambda}+d\boldsymbol{\lambda})}$:
\[\begin{align*} \braket{\psi(\boldsymbol{\lambda})|\psi(\boldsymbol{\lambda}+d\boldsymbol{\lambda})} &= \bra{\psi(\boldsymbol{\lambda})}\left(\ket{\psi(\boldsymbol{\lambda})}+\frac{\partial}{\partial\lambda_{\mu}}\ket{\psi(\boldsymbol{\lambda})}d\lambda^{\mu}+\frac{1}{2!}\frac{\partial}{\partial\lambda_{\mu}}\frac{\partial}{\partial\lambda_{\nu}}\ket{\psi(\boldsymbol{\lambda})}d\lambda^{\mu}d\lambda^{\nu}\right) \\ &= 1+iA_{\mu}(\boldsymbol{\lambda})d\lambda^{\mu}+\frac{1}{2}\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\mu}}\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})}d\lambda^{\mu}d\lambda^{\nu} \end{align*}\]Note that the derivative of the Berry connection is given by:
\[\begin{equation} \frac{\partial}{\partial\lambda_{\mu}}\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})} = \Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})} + \Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\mu}}\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})} \end{equation}\]which is equivalent to:
\[\begin{align*} \Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\mu}}\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})} &= \frac{\partial}{\partial\lambda_{\mu}}\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})} - \Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})}\\ &= \frac{\partial}{\partial\lambda_{\mu}}\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})}-(\gamma_{\mu\nu}-A_{\mu}(\boldsymbol{\lambda})A_{\nu}(\boldsymbol{\lambda})) \end{align*}\]Recall that the gauge invariant form of $\Braket{\frac{\partial}{\partial\lambda_{\mu}}\psi(\boldsymbol{\lambda})\vert\frac{\partial}{\partial\lambda_{\nu}}\psi(\boldsymbol{\lambda})}$ is given by $\gamma_{\mu\nu}-A_{\mu}(\boldsymbol{\lambda})A_{\nu}(\boldsymbol{\lambda})$. Using this relation, the third term of $\braket{\psi(\boldsymbol{\lambda})\vert\psi(\boldsymbol{\lambda}+d\boldsymbol{\lambda})}$ will be expressed as:
\[\begin{align*} \frac{1}{2}\Braket{\psi(\boldsymbol{\lambda})|\frac{\partial}{\partial\lambda_{\mu}}\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})}d\lambda^{\mu}d\lambda^{\nu} &= \frac{1}{2}\left[-(\gamma_{\mu\nu}-A_{\mu}(\boldsymbol{\lambda})A_{\nu}(\boldsymbol{\lambda})) + \frac{\partial}{\partial\lambda_{\mu}}\Braket{\psi\left(\boldsymbol{\lambda}\right)|\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})}\right]d\lambda^{\mu}d\lambda^{\nu} \\ &= -\frac{1}{2}(\gamma_{\mu\nu}-A_{\mu}(\boldsymbol{\lambda})A_{\nu}(\boldsymbol{\lambda})) d\lambda^{\mu}d\lambda^{\nu} + \frac{1}{2}\Braket{\psi\left(\boldsymbol{\lambda}\right)|\frac{\partial}{\partial\lambda_{\nu}}|\psi(\boldsymbol{\lambda})} d\lambda^{\nu} \\ &= -\frac{1}{2}(\gamma_{\mu\nu}-A_{\mu}(\boldsymbol{\lambda})A_{\nu}(\boldsymbol{\lambda})) d\lambda^{\mu}d\lambda^{\nu} - \frac{1}{2}iA_{\nu}(\boldsymbol{\lambda})d\lambda^{\nu} \end{align*}\]Therefore, $\braket{\psi(\boldsymbol{\lambda})\vert\psi(\boldsymbol{\lambda}+d\boldsymbol{\lambda})}$ is given by:
\[\begin{align*} \braket{\psi(\boldsymbol{\lambda})|\psi(\boldsymbol{\lambda}+d\boldsymbol{\lambda})} &= 1-\frac{1}{2}(\gamma_{\mu\nu}-A_{\mu}(\boldsymbol{\lambda})A_{\nu}(\boldsymbol{\lambda})) d\lambda^{\mu}d\lambda^{\nu} \\ &= 1-\frac{1}{2}g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu} \end{align*}\]whose square up to $\mathcal{O}(d\lambda^{2})$ gives rise to:
\[\begin{align*} ||\braket{\psi(\boldsymbol{\lambda})|\psi(\boldsymbol{\lambda}+d\boldsymbol{\lambda})}||^{2} &= \left|\left|1-\frac{1}{2}g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}\right|\right|^{2} \\ &= \left(1-\frac{1}{2}g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}\right)^{*}\left(1-\frac{1}{2}g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}\right) \\ &= \left(1-\frac{1}{2}g_{\nu\mu}d\lambda^{\mu}d\lambda^{\nu}\right)\left(1-\frac{1}{2}g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu}\right) \\ &= 1 - g_{\mu\nu}d\lambda^{\mu}d\lambda^{\nu} \end{align*}\]where one has used the symmetric relation of quantum metric; $g_{\mu\nu}=g_{\nu\mu}$. Note that the quantum metric $g_{\mu\nu}$ directly corresponds to the quantum distance $ds^{2}$ (or trace distance for pure states), which is given by:
\[\begin{equation} ds^{2} = 1 - ||\braket{\psi(\boldsymbol{\lambda})|\psi(\boldsymbol{\lambda}+d\boldsymbol{\lambda})}||^{2} = g_{\mu\nu}d\lambda^{\mu}\lambda^{\nu} \end{equation}\]When the state is given by a density matrix, the first term in the equation is called the Bures distance. The different concepts (quantum distance and Bures distance) become the same for a pure state only.
Example. Spin-$1/2$ system
A spin-$1/2$ spinor is represented as:
\[\begin{equation} \ket{\uparrow;\hat{\mathbf{n}}} = \begin{bmatrix} \cos\left(\frac{\theta}{2}\right) \\ \sin\left(\frac{\theta}{2}\right)e^{i\varphi} \end{bmatrix},~\ket{\downarrow;\hat{\mathbf{n}}} = \begin{bmatrix} -\sin\left(\frac{\theta}{2}\right) \\ \cos\left(\frac{\theta}{2}\right)e^{i\varphi} \end{bmatrix} \end{equation}\]where $\theta$ and $\varphi$ are the polar and azimuthal angles of $\hat{\mathbf{n}}$. The derivative of the spinor with respect to the polar and azimuthal angles reads:
\[\begin{align*} \ket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}} &= \begin{bmatrix} -\frac{1}{2}\sin\left(\frac{\theta}{2}\right) \\ \frac{1}{2}\cos\left(\frac{\theta}{2}\right)e^{i\varphi} \end{bmatrix},~\ket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}} = \begin{bmatrix} -\frac{1}{2}\cos\left(\frac{\theta}{2}\right) \\ -\frac{1}{2}\sin\left(\frac{\theta}{2}\right)e^{i\varphi} \end{bmatrix}, \\ \ket{\uparrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}} &= \begin{bmatrix} 0 \\ i\sin\left(\frac{\theta}{2}\right)e^{i\varphi} \end{bmatrix},~\ket{\downarrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}} = \begin{bmatrix} 0 \\ i\cos\left(\frac{\theta}{2}\right)e^{i\varphi} \end{bmatrix} \end{align*}\]Using these relations, one can obtain the quantum geometry tensor for $\mu=\theta$ and $\nu=\theta$:
\[\begin{align*} \mathfrak{G}^{\uparrow\uparrow}_{\theta\theta} &= \Braket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}}-\Braket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\uparrow;\hat{\mathbf{n}}}\Braket{\uparrow;\hat{\mathbf{n}}|\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}} \\ &= \frac{1}{4} - 0 = \frac{1}{4}, \\ \mathfrak{G}^{\uparrow\downarrow}_{\theta\theta} &= \Braket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}}-\Braket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\downarrow;\hat{\mathbf{n}}}\Braket{\uparrow;\hat{\mathbf{n}}|\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}} \\ &= 0-\left(+\frac{1}{2}\right)\left(-\frac{1}{2}\right) = \frac{1}{4}, \\ \mathfrak{G}^{\downarrow\uparrow}_{\theta\theta} &= \Braket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}}-\Braket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\uparrow;\hat{\mathbf{n}}}\Braket{\downarrow;\hat{\mathbf{n}}|\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}} \\ &= 0-\left(-\frac{1}{2}\right)\left(+\frac{1}{2}\right) = \frac{1}{4}, \\ \mathfrak{G}^{\downarrow\downarrow}_{\theta\theta} &= \Braket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}}-\Braket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\downarrow;\hat{\mathbf{n}}}\Braket{\downarrow;\hat{\mathbf{n}}|\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}} \\ &= \frac{1}{4} - 0 = \frac{1}{4} \\ \therefore \mathfrak{G}_{\theta\theta} &= 1 \end{align*}\]Likewise, for $\mu=\theta$ and $\nu=\varphi$,
\[\begin{align*} \mathfrak{G}^{\uparrow\uparrow}_{\theta\varphi} &= \Braket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\uparrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}}-\Braket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\uparrow;\hat{\mathbf{n}}}\Braket{\uparrow;\hat{\mathbf{n}}|\uparrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}} \\ &= \frac{1}{4} - 0 = \frac{1}{4}, \\ \mathfrak{G}^{\uparrow\downarrow}_{\theta\varphi} &= \Braket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\downarrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}}-\Braket{\uparrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\downarrow;\hat{\mathbf{n}}}\Braket{\uparrow;\hat{\mathbf{n}}|\downarrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}} \\ &= 0-\left(+\frac{1}{2}\right)\left(-\frac{1}{2}\right) = \frac{1}{4}, \\ \mathfrak{G}^{\downarrow\uparrow}_{\theta\varphi} &= \Braket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\uparrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}}-\Braket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\uparrow;\hat{\mathbf{n}}}\Braket{\downarrow;\hat{\mathbf{n}}|\uparrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}} \\ &= 0-\left(-\frac{1}{2}\right)\left(+\frac{1}{2}\right) = \frac{1}{4}, \\ \mathfrak{G}^{\downarrow\downarrow}_{\theta\varphi} &= \Braket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\downarrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}}-\Braket{\downarrow;\frac{\partial}{\partial\theta}\hat{\mathbf{n}}|\downarrow;\hat{\mathbf{n}}}\Braket{\downarrow;\hat{\mathbf{n}}|\downarrow;\frac{\partial}{\partial\varphi}\hat{\mathbf{n}}} \\ &= \frac{1}{4} - 0 = \frac{1}{4} \end{align*}\]The quantum metric and Berry curvature are written as:
\[\begin{align*} g_{\mu\nu} &= \begin{bmatrix} g_{\theta\theta} & g_{\theta\varphi} \\ g_{\varphi\theta} & g_{\varphi\varphi} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & \sin^{2}\left(\frac{\theta}{2}\right)\cos^{2}\left(\frac{\theta}{2}\right) \end{bmatrix}, \\ \Omega_{\mu\nu} &= \begin{bmatrix} \Omega_{\theta\theta} & \Omega_{\theta\varphi} \\ \Omega_{\varphi\theta} & \Omega_{\varphi\varphi} \end{bmatrix} = \begin{bmatrix} 0 & \sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right) \\ -\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right) & 0 \end{bmatrix} \end{align*}\]which implies that there are a nontrivial quantum metric and Berry curvature.
References
- Cheng, R. Quantum Geometric Tensor (Fubini-Study Metric) in Simple Quantum System: A pedagogical Introduction. arXiv (2010).