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1X F1⁄2C 1⁄4 2 qo r0+q2+ joj u X X X jCðq,oÞj2+ 0 C1C2C3C4d q d o (12) Quantum Critical Matter and Phase Transitions Chapter 280 301 energy (also sometimes called “action”) depending on the order parameter field C(x, t). For example, for an itinerant ferromagnet this leads to q 4Nbqioi where (q, o) are momentum and energy, the Fourier transform of (x, t); r0 is the tuning parameter that makes the system ferromagnetic or paramagnetic, and u0 is some effective order parameter interaction. Since this is a low- energy effective field theory, the correct way to study this is through renorma- lization group theory (Herbut, 2010; Wilson and Kogut, 1974; Yeomans, 1992) —a topic that goes beyond this brief review. Instead, we can infer the most important results from simple scaling arguments. A scaling argument uses that at the QPT, physical properties are the same at all length scales. Hence, if we multiply the momenta q and the energies o with some factor, the action Eq. (12) should remain invariant. The difference between the scaling of momenta and energies is characterized by the dynami- cal critical exponent z, recall Eq. (11). Simple inspection of the ferromagnetic action (12) implies z 1⁄4 3. Similarly, for antiferromagnets we find z 1⁄4 2 (Hertz, 1976). The upper critical dimension for this model is no longer 4, as was the case for the classical Ising model, but 4z with z the dynamical critical exponent. This implies that the ferromagnetic QPT in d>1 and the antiferromagnetic QPT in d>2 can be described by mean field exponents! The scaling arguments can be extended to nonzero temperatures. When- ever the inverse temperature b is smaller than the imaginary time correlation length xt, one expects that quantum critical fluctuations are dominant. This leads to the famous “quantum critical wedge,” T>ðggcÞnz, (13) the temperature regime where the system can be described as “quantum critical” and standard lore about interacting fermions no longer applies. In this regime, a scaling ansatz for the free energy can be formulated (Zhu et al., 2003), $f ðTx Þ$jgg jðd+zÞn $f T (14) t c jggcjnz from which we can directly infer that the low-temperature specific heat scales as FðTÞ$ 1 xdx t cVðT, g1⁄4gcÞ1⁄4T@2F $Td=z: @T2 (15) This is notably different from the standard FL result of a linear specific heat cV(T) $ gT where g is the Sommerfeld coefficient. That is why the system inside the quantum critical wedge is often referred to as a “NFL.”PDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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