Random transverse-field Ising model

To illustrate the rich behavior of quantum phase transitions in disordered systems, we now consider the random transverse-field Ising model, a random version of our first example. It is given by the Hamiltonian 

Exponent Generic Transition Multicritical Point Percolation Transition 

In dimensions d 1, an exact solution is not available because the strong disorder renormalization group can be implemented only numerically. Moreover, mapping the spin system onto free fermions is restricted to one dimension. Therefore, simulation studies have mostly used Monte Carlo methods. The quantum-to-classical mapping for the Hamiltonian in Eq. [30] can be performed analogously to the clean case. The result is a disordered classical Ising model in d + 1 dimensions with the disorder perfectly correlated in one dimension (in 1-1-1 dimensions, this is the famous McCoy-Wu modeF ). The classical Hamiltonian reads 

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