Is Chern number exactly an integer of half-integer?

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In summary, the question of whether the Chern number is exactly an integer or half-integer relates to the topological properties of certain quantum systems. Chern numbers, which arise in the study of topological phases of matter, can take on integer values associated with the quantized Hall effect. However, in specific contexts, such as systems with time-reversal symmetry, they may also yield half-integer values. The classification depends on factors like the symmetry of the system, the dimensionality, and the nature of the band structure, leading to a nuanced understanding of topological invariants in different physical scenarios.
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Mohammad-gl
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I'm trying to calculate the Chern number for a specific system. When I calculate the Chern number, it's not an integer, for example, I have 0.9 for it. Is it the wrong result or can I consider it as 1?
I'm trying to calculate the Chern number for a specific system. When I calculate the Chern number, it's not an integer, for example, I have 0.9 for it. Is it the wrong result or can I consider it as 1?
 
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What is a "Chem number"? Google just gives me the phone number of a company called "Chem Care".
 
  • #3
What is the system you are considering? In general, yes, it should be an integer.

You could have non-integer Chern numbers, for example, in a metallic system, because you do not have a gapped band structure at the Fermi level (Chern number non well defined in the conventional sense). But there are workarounds to still define an integer Chern number.

What methods are you using to calculate it? If it's just numerics you might adjust your code to get better results.
 
  • #4
I'm usi
dRic2 said:
What is the system you are considering? In general, yes, it should be an integer.

You could have non-integer Chern numbers, for example, in a metallic system, because you do not have a gapped band structure at the Fermi level (Chern number non well defined in the conventional sense). But there are workarounds to still define an integer Chern number.

What methods are you using to calculate it? If it's just numerics you might adjust your code to get better results.
I'm using tight-binding Hamiltonian, Kubo formula and MATLAB programming language. Is it ok if I calculate Chern number 1.93 or 0.7?
 
  • #5
Mohammad-gl said:
I'm usi

I'm using tight-binding Hamiltonian, Kubo formula and MATLAB programming language. Is it ok if I calculate Chern number 1.93 or 0.7?
Are you following a specific reference?
 
  • #6
I'm using this paper.
DOI: 10.1103/PhysRevB.106.205416

Can you help me with Matlab codes? If you can I will send you the code maybe I made a mistake in coding.
 

FAQ: Is Chern number exactly an integer of half-integer?

What is the Chern number?

The Chern number is a topological invariant used in mathematics and physics, particularly in the study of gauge theories and condensed matter systems. It quantifies the global properties of a vector bundle over a manifold, often associated with the behavior of wave functions in a system with a non-trivial topology.

Can the Chern number be a half-integer?

No, the Chern number is always an integer. It arises from the integration of a certain topological quantity (the first Chern form) over a two-dimensional manifold, which leads to quantized values. This quantization is a consequence of the underlying topology of the system.

Why is the Chern number important in physics?

The Chern number plays a crucial role in understanding various physical phenomena, such as the quantum Hall effect and topological insulators. It helps classify different phases of matter and can indicate the presence of edge states, which are protected by the topology of the system.

How is the Chern number calculated?

The Chern number can be calculated using the formula that involves integrating the curvature of a connection over the Brillouin zone or the parameter space of the system. Specifically, it is computed as an integral of the first Chern form, which is derived from the Berry connection associated with the eigenstates of the system.

Are there any exceptions or special cases regarding the Chern number?

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