Berry connection and curvature

In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. The concept was first introduced by S. Pancharatnam^[1] as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984^[2] emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.

In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian ${\displaystyle H(\mathbf {R} )}$ depends on a (vector) parameter ${\displaystyle \mathbf {R} }$ that varies with time ${\displaystyle t}$. If the ${\displaystyle n}$'th eigenvalue ${\displaystyle \varepsilon _{n}(\mathbf {R} )}$ remains non-degenerate everywhere along the path and the variation with time t is sufficiently slow, then a system initially in the normalized eigenstate ${\displaystyle |n(\mathbf {R} (0))\rangle }$ will remain in an instantaneous eigenstate ${\displaystyle |n(\mathbf {R} (t))\rangle }$ of the Hamiltonian ${\displaystyle H(\mathbf {R} (t))}$, up to a phase, throughout the process. Regarding the phase, the state at time t can be written as^[3] $${\displaystyle |\Psi _{n}(t)\rangle =e^{i\gamma _{n}(t)}\,e^{-{i \over \hbar }\int _{0}^{t}dt'\varepsilon _{n}(\mathbf {R} (t'))}\,|n(\mathbf {R} (t))\rangle ,}$$ where the second exponential term is the "dynamic phase factor." The first exponential term is the geometric term, with ${\displaystyle \gamma _{n}}$ being the Berry phase. From the requirement that the state ${\displaystyle |\Psi _{n}(t)\rangle }$ satisfies the time-dependent Schrödinger equation, it can be shown that $${\displaystyle \gamma _{n}(t)=i\int _{0}^{t}dt'\,\langle n(\mathbf {R} (t'))|{d \over dt'}|n(\mathbf {R} (t'))\rangle =i\int _{\mathbf {R} (0)}^{\mathbf {R} (t)}d\mathbf {R} \,\langle n(\mathbf {R} )|\nabla _{\mathbf {R} }|n(\mathbf {R} )\rangle ,}$$ indicating that the Berry phase only depends on the path in the parameter space, not on the rate at which the path is traversed.

In the case of a cyclic evolution around a closed path ${\displaystyle {\mathcal {C}}}$ such that ${\displaystyle \mathbf {R} (T)=\mathbf {R} (0)}$, the closed-path Berry phase is $${\displaystyle \gamma _{n}=i\oint _{\mathcal {C}}d\mathbf {R} \,\langle n(\mathbf {R} )|\nabla _{\mathbf {R} }|n(\mathbf {R} )\rangle .}$$ An example of physical systems where an electron moves along a closed path is cyclotron motion (details are given in the page of Berry phase). Berry phase must be considered to obtain the correct quantization condition.

A gauge transformation can be performed $${\displaystyle |{\tilde {n}}(\mathbf {R} )\rangle =e^{-i\beta (\mathbf {R} )}|n(\mathbf {R} )\rangle }$$ to a new set of states that differ from the original ones only by an ${\displaystyle \mathbf {R} }$-dependent phase factor. This modifies the open-path Berry phase to be ${\displaystyle {\tilde {\gamma }}_{n}(t)=\gamma _{n}(t)+\beta (t)-\beta (0)}$. For a closed path, continuity requires that ${\displaystyle \beta (T)-\beta (0)=2\pi m}$ (${\displaystyle m}$ an integer), and it follows that ${\displaystyle \gamma _{n}}$ is invariant, modulo ${\displaystyle 2\pi }$, under an arbitrary gauge transformation.

The closed-path Berry phase defined above can be expressed as $${\displaystyle \gamma _{n}=\int _{\mathcal {C}}d\mathbf {R} \cdot {\mathcal {A}}_{n}(\mathbf {R} )}$$ where $${\displaystyle {\mathcal {A}}_{n}(\mathbf {R} )=i\langle n(\mathbf {R} )|\nabla _{\mathbf {R} }|n(\mathbf {R} )\rangle }$$ is a vector-valued function known as the Berry connection (or Berry potential). The Berry connection is gauge-dependent, transforming as ${\displaystyle {\tilde {\mathcal {A}}}_{n}(\mathbf {R} )={\mathcal {A}}_{n}(\mathbf {R} )+\nabla _{\mathbf {R} \,}\beta (\mathbf {R} )}$. Hence the local Berry connection ${\displaystyle {\mathcal {A}}_{n}(\mathbf {R} )}$ can never be physically observable. However, its integral along a closed path, the Berry phase ${\displaystyle \gamma _{n}}$, is gauge-invariant up to an integer multiple of ${\displaystyle 2\pi }$. Thus, ${\displaystyle e^{i\gamma _{n}}}$ is absolutely gauge-invariant, and may be related to physical observables.

The Berry curvature is an anti-symmetric second-rank tensor derived from the Berry connection via $${\displaystyle \Omega _{n,\mu \nu }(\mathbf {R} )={\partial \over \partial R^{\mu }}{\mathcal {A}}_{n,\nu }(\mathbf {R} )-{\partial \over \partial R^{\nu }}{\mathcal {A}}_{n,\mu }(\mathbf {R} ).}$$ In a three-dimensional parameter space the Berry curvature can be written in the pseudovector form $${\displaystyle \mathbf {\Omega } _{n}(\mathbf {R} )=\nabla _{\mathbf {R} }\times {\mathcal {A}}_{n}(\mathbf {R} ).}$$ The tensor and pseudovector forms of the Berry curvature are related to each other through the Levi-Civita antisymmetric tensor as ${\displaystyle \Omega _{n,\mu \nu }=\epsilon _{\mu \nu \xi }\,\mathbf {\Omega } _{n,\xi }}$. In contrast to the Berry connection, which is physical only after integrating around a closed path, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space, and has proven to be an essential physical ingredient for understanding a variety of electronic properties.^[4][5]

For a closed path ${\displaystyle {\mathcal {C}}}$ that forms the boundary of a surface ${\displaystyle {\mathcal {S}}}$, the closed-path Berry phase can be rewritten using Stokes' theorem as $${\displaystyle \gamma _{n}=\int _{\mathcal {S}}d\mathbf {S} \cdot \mathbf {\Omega } _{n}(\mathbf {R} ).}$$ If the surface is a closed manifold, the boundary term vanishes, but the indeterminacy of the boundary term modulo ${\displaystyle 2\pi }$ manifests itself in the Chern theorem, which states that the integral of the Berry curvature over a closed manifold is quantized in units of ${\displaystyle 2\pi }$. This number is the so-called Chern number, and is essential for understanding various quantization effects.

Finally, by using ${\displaystyle \left\langle n|\partial H/\partial \mathbf {R} |n'\right\rangle =\left\langle \partial n/\partial \mathbf {R} |n'\right\rangle (\varepsilon _{n}-\varepsilon _{n'})}$ for ${\displaystyle n\neq n'}$, the Berry curvature can also be written as a summation over all the other eigenstates in the form $${\displaystyle \Omega _{n,\mu \nu }(\mathbf {R} )=i\sum _{n'\neq n}{\frac {1}{(\varepsilon _{n}-\varepsilon _{n'})^{2}}}{\left(\left\langle n\right|{\frac {\partial H}{\partial R_{\mu }}}\left|n'\right\rangle \left\langle n'\right|{\frac {\partial H}{\partial R_{\nu }}}\left|n\right\rangle -\left\langle n\right|{\frac {\partial H}{\partial R_{\nu }}}\left|n'\right\rangle \left\langle n'\right|{\frac {\partial H}{\partial R_{\mu }}}\left|n\right\rangle \right)}.}$$ Note that the curvature of the nth energy level is contributed by all the other energy levels. That is, the Berry curvature can be viewed as the result of the residual interaction of those projected-out eigenstates. ^[5] This gives the local conservation law for the Berry curvature, ${\displaystyle \sum _{n}\Omega _{n,\mu \nu }(\mathbf {R} )=0,}$ if we sum over all possible energy levels for each value of ${\displaystyle \mathbf {R} .}$ This equation also offers the advantage that no differentiation on the eigenstates is involved, and thus it can be computed under any gauge choice.

The Hamiltonian of a spin-1/2 particle in a magnetic field can be written as^[3] $${\displaystyle H=\mu \mathbf {\sigma } \cdot \mathbf {B} ,}$$ where ${\displaystyle \mathbf {\sigma } }$ denote the Pauli matrices, ${\displaystyle \mu }$ is the magnetic moment, and B is the magnetic field. In three dimensions, the eigenstates have energies ${\displaystyle \pm \mu B}$ and their eigenvectors are $${\displaystyle |u_{-}\rangle ={\begin{pmatrix}\sin {\theta \over 2}e^{-i\phi }\\-\cos {\theta \over 2}\end{pmatrix}},|u_{+}\rangle ={\begin{pmatrix}\cos {\theta \over 2}e^{-i\phi }\\\sin {\theta \over 2}\end{pmatrix}}.}$$ Now consider the ${\displaystyle |u_{-}\rangle }$ state. Its Berry connection can be computed as ${\textstyle {\mathcal {A}}_{\theta }=\langle u_{-}|i{\frac {1}{r}}\partial _{\theta }|u_{-}\rangle =0,}$ ${\displaystyle {\mathcal {A}}_{\phi }=\langle u_{-}|i{\tfrac {1}{r\sin {\theta }}}\partial _{\phi }|u_{-}\rangle ={\frac {\sin ^{2}{\theta \over 2}}{r\sin {\theta }}}}$, and the Berry curvature is ${\displaystyle \ \Omega _{r}={\frac {1}{r\sin {\theta }}}[\partial _{\theta }({\mathcal {A}}_{\phi }\sin {\theta })-\partial _{\phi }{\mathcal {A}}_{\theta }]{\hat {r}}={\frac {1}{2r^{2}}}{\hat {r}}.}$ If we choose a new gauge by multiplying ${\displaystyle |u_{-}\rangle }$ by ${\displaystyle e^{i\phi }}$ (or any other phase ${\displaystyle e^{i\alpha \phi }}$, ${\displaystyle \alpha \in \mathbb {R} }$), the Berry connections are ${\displaystyle {\mathcal {A}}_{\theta }=0}$ and ${\displaystyle {\mathcal {A}}_{\phi }=-{\frac {\cos ^{2}{\theta \over 2}}{r\sin {\theta }}}}$, while the Berry curvature remains the same. This is consistent with the conclusion that the Berry connection is gauge-dependent while the Berry curvature is not.

The Berry curvature per solid angle is given by ${\displaystyle {\overline {\Omega }}_{\theta \phi }=\Omega _{\theta \phi }/\sin \theta =1/2}$. In this case, the Berry phase corresponding to any given path on the unit sphere ${\displaystyle {\mathcal {S}}^{2}}$ in magnetic-field space is just half the solid angle subtended by the path. The integral of the Berry curvature over the whole sphere is therefore exactly ${\displaystyle 2\pi }$, so that the Chern number is unity, consistent with the Chern theorem.

The Berry phase plays an important role in modern investigations of electronic properties in crystalline solids^[5] and in the theory of the quantum Hall effect.^[6] The periodicity of the crystalline potential allows the application of the Bloch theorem, which states that the Hamiltonian eigenstates take the form $${\displaystyle \psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} ),}$$ where ${\displaystyle n}$ is a band index, ${\displaystyle \mathbf {k} }$ is a wavevector in the reciprocal-space (Brillouin zone), and ${\displaystyle u_{n\mathbf {k} }(\mathbf {r} )}$ is a periodic function of ${\displaystyle \mathbf {r} }$. Due to translational symmetry, the momentum operator ${\displaystyle {\hat {p}}_{i}}$ could be replaced with ${\displaystyle {\frac {m}{\hbar }}\partial H/\partial k_{i}}$ by the Peierls substitution and the wavevector ${\displaystyle \mathbf {k} }$ plays the role of the parameter ${\displaystyle \mathbf {R} }$.^[5] Thus, one can define Berry phases, connections, and curvatures in the reciprocal space. For example, in an N-band system, the Berry connection of the nth band in reciprocal space is $${\displaystyle {\mathcal {A}}_{n}(\mathbf {k} )=i\langle u_{n\mathbf {k} }|\nabla _{\mathbf {k} }|u_{n\mathbf {k} }\rangle .}$$ In the system, the Berry curvature of the nth band ${\displaystyle \Omega _{n,\mu \nu }(\mathbf {k} )}$ is given by all the other N − 1 bands for each value of ${\displaystyle \mathbf {k} .}$ In a 2D crystal, the Berry curvature only has the component out of the plane and behaves as a pseudoscalar. It is because there only exists in-plane translational symmetry when translational symmetry is broken along z direction for a 2D crystal. Because the Bloch theorem also implies that the reciprocal space itself is closed, with the Brillouin zone having the topology of a 3-torus in three dimensions, the requirements of integrating over a closed loop or manifold can easily be satisfied. In this way, such properties as the electric polarization, orbital magnetization, anomalous Hall conductivity, and orbital magnetoelectric coupling can be expressed in terms of Berry phases, connections, and curvatures.^[5][7][8]

• The quantum phase, five years after. by M. Berry.
• Berry Phases and Curvatures in Electronic Structure Theory A talk by D. Vanderbilt.
• Berry-ology, Orbital Magnetolectric Effects, and Topological Insulators - A talk by D. Vanderbilt.