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Berry curvature is defined by

$$\Omega_{n,\mu\nu}(R) = \frac{\partial A_{n,\nu}}{\partial R^{\mu}} - \frac{\partial A_{n,\mu}}{\partial R^{\nu}} $$ where $R$ is an parameter in hamiltonian. I think it is well defined if the dimension of parameter $R$ is more than 2. But what if the dimension of parameter $R$ is 1? Is it possible to say something about Berry curvature if the dimension of parameter is 1? If the dimension is 1, I think even the closed path integral for Berry phase is not well derfined.

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If the dimension of $R=1$, them $\mu=\nu=$ $\alpha$. The $\Omega_{n,\alpha,\alpha}=\partial_{\alpha}A_{n,\alpha}-\partial_{\alpha}A_{n,\alpha}=0$. On the other hand, a closed path, can be defined as $R(\alpha (t))=R(\alpha(t+T))$. So, I don't think any problem in defining the closed path in one parameter case, as a close path depends on the initial and final point.

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  • $\begingroup$ If $\Omega_{n,\nu\mu}=0$, Berry phase is always zero in 1 dimensional parameter space. But it can have any non-zero values in higher dimensions. What's the difference between one and higher dimension? $\endgroup$ – user42298 Feb 28 '17 at 11:50

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