Critical Exponents for Quantum Phase Transitions

[ianyappy]

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 :)

 

[azntoon]

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!