A What are the methods for calculating quantum phase transition?

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The discussion centers on the mathematical methods used to calculate quantum phase transitions. Participants emphasize the importance of having a clearly defined problem to facilitate understanding and calculations. Various mathematical approaches are suggested, but specifics are lacking without context. Clarification on diagrams and their meanings is also requested to enhance comprehension. Overall, a focused question is deemed essential for progressing in the study of quantum phase transitions.
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Trying to understand what the math methods are exist to determine quantum phase transition and how I can calculate it, by what methods.
 
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I don't know what you are talking about and what the meaning of your diagram is. Do you have a concrete physical problem in mind? Asking a clearly defined question is usually more than half way to a solution!
 
For the quantum state ##|l,m\rangle= |2,0\rangle## the z-component of angular momentum is zero and ##|L^2|=6 \hbar^2##. According to uncertainty it is impossible to determine the values of ##L_x, L_y, L_z## simultaneously. However, we know that ##L_x## and ## L_y##, like ##L_z##, get the values ##(-2,-1,0,1,2) \hbar##. In other words, for the state ##|2,0\rangle## we have ##\vec{L}=(L_x, L_y,0)## with ##L_x## and ## L_y## one of the values ##(-2,-1,0,1,2) \hbar##. But none of these...

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