content_image/363475.jpg content_image/363476.jpg content_image/363477.jpg content_image/363484.jpg
matrix, 

where 

and 

<img src='content_image/363475.jpg'>

<img src='content_image/363476.jpg'>

<img src='content_image/363477.jpg'>

With the above Hamiltonian (7), it is straightforward to obtain the Hofstadter butterfly-like energy spectrum of the kagom´ e lattice, which is shown in Fig. 2. 

From Fig. 2, one can easily observe three symmetries of this Hofstadter butterfly-like spectrum of the kagom´ e lattice [4]: (i) The spectrum at f is the same as that at f + j with j any integer; (ii) The spectrum is also unchanged on changing f to − f , because if there is an eigenstate with energy ϵ for field f , then the corresponding complex conjugate of this eigenstate is also an eigenstate with the same energy ϵ for field − f ; (iii) The spectrum is inverted when the field f changes to f + 1 / 2. Thus the highest-energy states for field near f =1 / 2 are equivalent to the lowest-energy states near zero field. 

## III. THE PROPERTIES OF THE ORBITAL MAGNETIZATION 

To obtain the orbital magnetization of the kagom´ e lattice in an external magnetic field and with finite temperature, let us first write down the single-particle free energy as follows [16]: 

<img src='content_image/363484.jpg'>

Here, Ω$_{n}$( k ) is the Berry curvature of electronic Bloch states defined by Ω$_{n}$( k )= i 〈∇ $_{k}$u$_{n}$$_{k}$ | × |∇ $_{k}$u$_{n}$$_{k}$ 〉 with | u$_{n}$$_{k}$ 〉 being the periodic part of Bloch wave for the n th band. Its 

