Pierels substitution integral approximation

In summary, the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes discusses the Peierls substitution for including an external field in a tight binding model. In equation 3.9a, they make an approximation using the vector potential A(s,t) that does not vary widely over the integration path. However, in equation 3.10, the factor of 1/2 is missing which is a mistake according to basic calculus. This error is present in the textbook.
  • #1
DeathbyGreen
84
16
In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (page 17) they are discussing including an external field in a tight binding model, the Peierls substitution. They make the statement that "if the vector potential" A(s,t) "does not vary widely over the integration path" (when moving from lattice site R to R') we can use the approximation

[itex] \int_R^{R'}A(s,t)\cdot ds \approx (R-R')\cdot \frac{1}{2}(A(R',t)+A(R,t)) \approx. (R'-R)\cdot \frac{1}{2}A\left(\frac{R'+R}{2},t\right)[/itex]

which is equation (3.9a). In equation 3.10, they use this substitution (changing variables with r=i-j):

[itex]
H_{ext} = \sum_{k_1k_2\alpha\beta}c^{\dagger}_{k_1\alpha}c_{k_2\beta}\frac{1}{N}\sum_{rj}e^{i(k_2-k_1)j - ik_1r}h^{\alpha\beta}_r(t)i\int_j^{j+r}A_p(t)dl\\
\approx
\sum_{k_1k_2\alpha\beta}c^{\dagger}_{k_1\alpha}c_{k_2\beta}\frac{1}{N}\sum_{rj}e^{i(k_2-k_1)j - ik_1r}h^{\alpha\beta}_r(t)irA_p(j
+\frac{r}{2},t)
[/itex]

I don't understand where the [itex] \frac{1}{2}[/itex] goes. It seems to disappear going from equation 3.9a to 3.10.
 
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  • #2
DeathbyGreen said:
In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (page 17) they are discussing including an external field in a tight binding model, the Peierls substitution. They make the statement that "if the vector potential" A(s,t) "does not vary widely over the integration path" (when moving from lattice site R to R') we can use the approximation

[itex] \int_R^{R'}A(s,t)\cdot ds \approx (R-R')\cdot \frac{1}{2}(A(R',t)+A(R,t)) \approx. (R'-R)\cdot \frac{1}{2}A\left(\frac{R'+R}{2},t\right)[/itex]
The factor 1/2 you have in the last expression on the right should not be there. (## A(s,t) ## is simply getting evaluated at the midpoint of the interval.)
 
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  • #3
Thank you! Do you have a source for that? The equation I wrote

[itex] \int_R^{R'}A(s,t)\cdot ds \approx (R-R')\cdot \frac{1}{2}(A(R',t)+A(R,t)) \approx. (R'-R)\cdot \frac{1}{2}A\left(\frac{R'+R}{2},t\right)[/itex]

Is straight from the textbook, no typos.
 
  • #4
DeathbyGreen said:
Thank you! Do you have a source for that? The equation I wrote

[itex] \int_R^{R'}A(s,t)\cdot ds \approx (R-R')\cdot \frac{1}{2}(A(R',t)+A(R,t)) \approx. (R'-R)\cdot \frac{1}{2}A\left(\frac{R'+R}{2},t\right)[/itex]

Is straight from the textbook, no typos.
Then they goofed. This is basic calculus. The 1/2 in the 3rd expression (in front of the ## A ##), is in error.
 
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  • #5
That textbook is full of errors. They did a really bad job.
 
  • #6
From my lecturers and students in QFT,Electromagnetism, Statistical Mechanics and Solid State Physics every expression should be regarded as "upto multiple numerical factor" correct.

In your case it's only half, that's superb! :-D
 

FAQ: Pierels substitution integral approximation

1. What is Pierels substitution integral approximation?

Pierels substitution integral approximation is a mathematical method used to approximate the value of an integral. It involves substituting a new variable into the integral and using it to simplify the expression. This method is commonly used in cases where the integral cannot be solved using traditional methods, such as when the integrand is too complex or when the limits of integration are infinite.

2. How does Pierels substitution integral approximation work?

The method involves substituting a new variable, usually denoted as u, into the integral. This new variable is chosen in such a way that it simplifies the expression and makes it easier to integrate. The integral is then rewritten in terms of u and solved using traditional integration techniques. The final step is to substitute back the original variable to obtain an approximate value of the integral.

3. When should Pierels substitution integral approximation be used?

Pierels substitution integral approximation should be used when traditional integration methods are not applicable or when the integral is too complex to be solved by hand. It is also useful when the limits of integration are infinite or when the integrand involves trigonometric functions or other complicated expressions.

4. What are the limitations of Pierels substitution integral approximation?

One of the main limitations of Pierels substitution integral approximation is that it only provides an approximate value of the integral, not the exact value. The accuracy of the approximation depends on the choice of the substitution variable and the complexity of the integrand. Additionally, this method may not work for all types of integrals, and it may require a lot of algebraic manipulation to obtain the final result.

5. Are there any alternatives to Pierels substitution integral approximation?

Yes, there are other methods for approximating integrals, such as the trapezoidal rule, Simpson's rule, and the midpoint rule. These methods involve dividing the interval of integration into smaller subintervals and approximating the integral using the area under a curve.

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