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Quantum Critical Matter and Phase Transitions Chapter 280 299 out the correlation functions for an inhomogeneous order parameter m(x) yields n1⁄41/2 and 1⁄40. Finally, an external magnetic field can be included by adding hRddxmðxÞ to the free energy, which yields d1⁄43 and g1⁄41. (See chapter 4 from Yeomans, 1992 for more details.) Mean field theory ignores fluctuations, but with increasing dimensionality of the system these fluctuations become less and less relevant. In fact, the mean field exponents become exact in d>2a dimensions. This is called n the upper critical dimension, so that, for example, the Ising model in d>4 dimensions is determined by mean field exponents. We are now in a position to continue toward quantum systems, where the phase transition does not occur as a function of temperature but at zero tem- perature due to a change in model parameters. Because in a quantum system the Hamiltonian is an operator the partition function is now given by, Z 1⁄4 Tr exp bH^: (7) The Ising model can be extended to a quantum model, known as the trans- verse field Ising model. The properties of this model are described in detail by Sachdev (2011). It is the prime example of a magnetic system exhibiting a QPT. The Hamiltonian is XX HTFIM 1⁄4Jg Sxi J SziSzj i hiji (8) where J>0 is the magnetic exchange coupling between spins, and g is the magnitude of a transversal magnetic field. Now g acts as the tuning parameter that can induce a phase transition between a ferromagnetic phase and a para- magnetic phase. For g≪1 (ferro)magnetic order prevails with two possible ground states, YY jc0i1⁄4 j"ii, or j#ii: (9) ii Oppositely, for a large transverse field g≫1, the ground state is an uncorre- lated (hSzi Szj i 1⁄4 dij ) paramagnet Y Y pffiffi jc0i1⁄4 j!ii 1⁄4 ðj"ii +j#iiÞ= 2: ii The paramagnet and the ferromagnet are distinctively different. Therefore, under increase of the transverse field g there will be a QPT from the ferro- to the paramagnet at g1⁄4gc. Suzuki (1976) discovered that the free energy of the d-dimensional quan- tum Ising model is equivalent to the (d+1)-dimensional classical Ising model. Explicitly, the quantum partition function can be rewritten as a Feynman path integral where the inverse temperature b1⁄41/kBT acts an extra dimensionPDF Image | HANDBOOK ON THE PHYSICS AND CHEMISTRY OF RARE EARTHS
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