Critical Exponents for Quantum Phase Transitions

In summary, the conversation discusses the change in relations between critical exponents for a quantum phase transition (QPT). Some sources state that at zero temperature, a quantum system is mapped to a classical system in one higher dimension, while others mention the inclusion of a dynamic critical exponent in finite-size scaling laws. These statements are not contradictory, as the Mermin-Wagner theorem explains that at zero temperature, the critical exponents for the QPT are the same as those of a classical phase transition in one higher dimension. However, when considering finite-size scaling, the universality class of the QPT is in a D+z classical system, rather than a D+1 system.
  • #1
ianyappy
12
0
Hi All, I'm doing an undergraduate project regarding QPTs for some variation of a AF Heisenberg hamiltonian. I'm a little confused about the change in relations between the critical exponents for a QPT. Some books/papers state that the quantum system in D dimensions is mapped to a classical D + 1 dimension system at T = 0. Yet they also say that in the finite-size scaling laws, we have to change D to D + z, where z is the dynamic critical exponent. So does that mean that the universality class of the QPT is in Some D + z classical system, but beyond the critical parameter, it appears like some D + 1 classical system? Is there some kind of contradiction between these two statements, or am I missing something? I'm afraid I'm not entirely familiar with the subject so I kind of assume universality class and mapping to a new classical system are kind of the same thing. Would appreciate any help with thanks :)
 
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  • #2
The statements you mention are actually not contradictory. The mapping from a quantum system to a classical system in one higher dimension refers to the fact that at zero temperature, quantum fluctuations are completely suppressed and the critical exponents characterizing the quantum phase transition are exactly the same as those of a classical phase transition in one higher dimension. This is known as the Mermin-Wagner theorem. However, when considering finite-size scaling in a quantum system, dynamic critical exponents become relevant, which means that the universality class of the quantum phase transition is in some D+z classical system, rather than a D+1 system. Hope this helps!
 

1. What are critical exponents for quantum phase transitions?

Critical exponents for quantum phase transitions are mathematical values used to describe the behavior of a physical system at the point of a phase transition, where the system undergoes a sudden change in its properties. These exponents are specific to quantum systems and are related to the scaling behavior of physical quantities near the transition point.

2. How are critical exponents determined?

Critical exponents are determined through experimental measurements and theoretical calculations. In experiments, physical quantities such as specific heat or magnetic susceptibility are measured at different temperatures near the transition point. In theory, critical exponents are obtained from mathematical models that describe the behavior of the system at the transition point.

3. What is the significance of critical exponents?

Critical exponents provide important insights into the behavior of physical systems at the point of phase transitions. They can help scientists understand the universal properties of different quantum systems, such as the Ising model or the XY model. They also play a crucial role in the development of theories and models to describe phase transitions and critical phenomena.

4. How do critical exponents differ from classical phase transitions?

Critical exponents for quantum phase transitions differ from those for classical phase transitions in several ways. In quantum systems, critical exponents can take on non-integer values, whereas in classical systems they are typically integer values. Additionally, quantum critical points are characterized by a breakdown in the Landau theory of phase transitions, while classical phase transitions can be described by this theory.

5. How do critical exponents relate to the renormalization group?

The renormalization group is a powerful mathematical tool used to study critical phenomena, including quantum phase transitions. Critical exponents play a key role in the renormalization group approach, as they determine the scaling behavior of physical quantities near the transition point. By studying the behavior of critical exponents, scientists can gain a deeper understanding of the underlying physics of quantum phase transitions and their universal properties.

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