A Can I change topology of the physical system smoothly?

[lichen1983312]

I am encountering this kind of problem in physics. The problem is like this:
Some quantity $A$ is identified as a potential field of a $U(1)$ bundle on a space $M$ (usually a torus), because it transforms like this ${A_j}(p) = {A_i}(p) + id\Lambda (p)$ in the intersection between patches ${U_i} \cap {U_j}$. So there should be a $U(1)$ bundle corresponding to this potential field and a Chern number is defined to characterize this bundle (or the physical system)
$c = \frac{i}{{2\pi }}\int_T F$
Where $F$ is the field strength $F = dA$.

One may change the physical system smoothly so that the potential field also changes smoothly. Let's say the system is changing with $t$, and the potential field becomes a smooth function $A(t)$, while the transform rule ${A_j}(p) = {A_i}(p) + id\Lambda (p)$ is still valid.

So the question is, is this smooth change of system possible to change the Chern number $c$ ?

 

[lavinia]

I am encountering this kind of problem in physics. The problem is like this:
Some quantity $A$ is identified as a potential field of a $U(1)$ bundle on a space $M$ (usually a torus), because it transforms like this ${A_j}(p) = {A_i}(p) + id\Lambda (p)$ in the intersection between patches ${U_i} \cap {U_j}$. So there should be a $U(1)$ bundle corresponding to this potential field and a Chern number is defined to characterize this bundle (or the physical system)
$c = \frac{i}{{2\pi }}\int_T F$
Where $F$ is the field strength $F = dA$.

One may change the physical system smoothly so that the potential field also changes smoothly. Let's say the system is changing with $t$, and the potential field becomes a smooth function $A(t)$, while the transform rule ${A_j}(p) = {A_i}(p) + id\Lambda (p)$ is still valid.

So the question is, is this smooth change of system possible to change the Chern number $c$ ?

I think you are changing the connection (gauge field) not the bundle. The Chern number is an invariant of the bundle. It is the same for any connection. Although the Chern form will be different its integral over the surface will be the same.

 

[lichen1983312]

I think you are changing the connection (gauge field) not the bundle. The Chern number is an invariant of the bundle. It is the same for any connection. Although the Chern form will be different its integral over the surface will be the same.

Is this possible? let's say there are two principal bundles ${P_1}$ and ${P_2}$ over $M$, and ${A_1}$ and ${A_2}$ are corresponding gauge fields. Is it possible to smoothly change ${A_1}$ into ${A_2}$?

 

[lichen1983312]

Maybe the question is not so clear, I will use an example.

A physical system is represented by k-dependent Hamiltonian operators $\hat H(k)$, where $k$ is a point in the torus. Each linear operator $\hat H(k)$ has a set of discrete eigen-functions such that $\hat H(k)\left| {{u_n}(k,r)} \right\rangle = {E_n}(k)\left| {{u_n}(k,r)} \right\rangle$, where ${E_n}(k)$ are energy eigen-values and are real. $\left| {{u_n}(k,r)} \right\rangle$ is a smooth function over $k$ and the Cartesian coordinates $r$. $\left| {{u_n}(k,r)} \right\rangle$ is required to be normalized such that $\left\langle {{u_m}(k,r)} \right.\left| {{u_n}(k,r)} \right\rangle = {\delta _{m,n}}$ where $\left\langle {{u_m}(k,r)} \right.\left| {{u_n}(k,r)} \right\rangle$ are defined as $\left\langle {{u_m}(k,r)} \right.\left| {{u_n}(k,r)} \right\rangle = \int {dr} \,u_m^ * (k,r){u_n}(k,r)$.

However there is a phase ambiguity of $\left| {{u_n}(k,r)} \right\rangle$, since
$\hat H(k){e^{i\Lambda (k)}}\left| {{u_n}(k,r)} \right\rangle = {e^{i\Lambda (k)}}\hat H(k)\left| {{u_n}(k,r)} \right\rangle = {E_n}(k){e^{i\Lambda (k)}}\left| {{u_n}(k,r)} \right\rangle$
So $\left| {u{'_n}(k,r)} \right\rangle = {e^{i\Lambda (k)}}\left| {{u_n}(k,r)} \right\rangle$ is also smooth over $k$ and $r$, if ${\Lambda (k)}$ is smooth.
For simplicity, assume there is only one energy eigen state. Now define a special connection, called the Berry's connection as
$A = \left\langle {u(k,r)} \right|{d_p}\left| {u(k,r)} \right\rangle = \left\langle {u(k,r)} \right|\frac{\partial }{{\partial {k^\mu }}}\left| {u(k,r)} \right\rangle d{k^\mu }$
($\left\langle {u(k,r)} \right|\frac{\partial }{{\partial {k^\mu }}}\left| {u(k,r)} \right\rangle$ is pure imaginary)
so if let $\left| {u(k,r)} \right\rangle \to \left| {u'(k,r)} \right\rangle$
$A \to A' = A + id\Lambda$

In the case of the graph used here, we may define smooth potential fields ${A^A}$ and ${A^B}$ on patches A and B. on the boundary set the transition function as ${t^{AB}} = {e^{i\Lambda (k)}}$ , using $F = dA$, the Chern number is
$c = \frac{i}{{2\pi }}\int_T F$

Now if I let the Hamiltonian $\hat H(k)$ depend on a parameter $t$ in a smooth way, the energy eigen-state $\left| {{u_n}(k,r,t)} \right\rangle$as well as the Berry's connection $A(t)$ change with $t$ smoothly.

is it possible at certain $t$, the Chern number change its value ?