Magnetic quantum phase transitions

We describe here computational approaches to quantum phase transitions that rely on the quantum-to-classical mapping. The number of transitions that can be studied by these approaches is huge our discussion is therefore not meant to be comprehensive. Following an introduction to the method, we discuss a few representative examples, mostly from the area of magnetic quantum phase transitions. 

The first example we consider is arguably the simplest model displaying a quantum phase transition—the quantum Ising model in a transverse field. It can be viewed as a toy model for the magnetic quantum phase transition of LiHop4 discussed in the introductory section. For this system, we explain... 

The examples discussed so far are all magnetic quantum phase transitions. Our last example in this section on quantum-to-classical mapping is a quite different transition, viz. the superconductor-insulator transition in two-dimensional dirty boson systems. Experimentally, this transition can be realized in helium absorbed in a porous medium or in granular superconducting films as an example. 

Having considered several examples of magnetic quantum phase transitions, we now turn to the superfluid-insulator transition in many-boson systems. In the section on Classical Monte Carlo Approaches we discussed how the universal critical behavior of this transition can be determined by mapping the Bose-Hubbard model, Eq. [32], onto the classical (d -I-1)-dimensional link current Hamiltonian, Eq. [35], which can then be simulated using classical Monte Carlo methods. 

Fermi-Liquid Instabilities at Magnetic Quantum Phase Transitions. 

The examples of phase transitions mentioned above occur at nonzero temperature. At these so-called thermal or classical transitions, the ordered phase (the ice crystal or the ferromagnetic state of iron) is destroyed by thermal fluctuations. In the last two decades or so, considerable attention has focused on a very different class of phase transitions. These new transitions occur at zero temperature when a nonthermal parameter such as pressure, chemical composition, or magnetic field is changed. The fluctuations that destroy the ordered phase in these transitions cannot be of a thermal nature. Instead, they are quantum fluctuations that are a consequence of Heisenberg s uncertainty principle. For this reason, these zero-temperature transitions are called quantum phase transitions. 

In applying a large magnetic field of 0 - 70 Tesla SrCu (60.3)2 crosses over into new quantum phases observable as distinct plateaus in the magnetization [21-23], These plateaus correspond to 1/8,1/4 and 1/3 of the saturation magnetization of the spin system [24—26] and are understood as field induced triplet states that are divided in a commensurate way on the dimer lattice. Recent NMR experiments on the 1/8 plateau at 28 Tesla and at very low temperatures (35 mK) show a transition-like crystallization of triplets within a larger supercell and Friedel-like oscillations of the field induced spin polarization [27],... 

One of the most interesting aspects of energy transport is the excitation percolation transition (, and its similarity (10) to magnetic phase transitions and other critical phenomena (, 8). In its simplest form the problem is one of connectivity. In a binary system, made only of hosts and donors, the question is can the excitation travel from one side of the material to the other The implicit assumption is that there are excitation-transfer-bonds only between two donors that are "close enough", where "close enough" has a practical aspect (e.g. defined by the excitation transfer probability or time). Obviously, if there is a succession of excitation-bonds from one edge of the material to the other, one has "percolation", i.e. a connected chain of donors forming an excitation conduit. We note that the excitation-bonds seldom correspond to real chemical bonds rather more often they correspond to van-der-Walls type bonds and most often they correspond to a dipole-dipole or equivalent quantum-mechanical interaction. 

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