Transverse-field Ising model

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 transverse-field Ising model is defined on a d-dimensional hypercubic (i.e., square, cubic, etc.) lattice. Each site is occupied by a quantum spin-. The spins interact via a ferromagnetic nearest-neighbor exchange interaction / 0 between their z components. The transverse magnetic field couples to the x components of the spins. The Hamiltonian of the model is given by 

Because the 2 and x components of the spin operators do not commute, the partition function cannot be factorized simply into an interaction part and a transverse-field part. However, we can use the Trotter product formula,  

This representation of the partition function of the transverse-field Ising model is identical to the partition function of an anisotropic classical Ising model in d+1 dimensions with coupfing constants e/ in the d-space dimensions and K in the timelike direction. The classical Hamiltonian reads 

Because the interactions are short ranged (nearest neighbor only) in both space- and timelike directions, the anisotropy does not play a role in the critical behavior of this classical model. We thus conclude that the quantum phase transition of the d-dimensional quantum Ising model in a transverse field falls into the universality class of the (d - - l)-dimensional classical Ising model. This establishes the quantum-to-classical mapping (for a slightly different derivation based on transfer matrices see Ref. 10). 

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The two-dimensional transverse-field Ising model maps onto the three-dimensional classical Ising model, which is not exactly solvable. However, the critical behavior has been determined with high precision using Monte Carlo and series expansion methods (see, e.g.. Ref. 59). The exponent values are p 0.326, 1.247, v 0.629. The other exponents can be found... 

The dissipative transverse-field Ising chain consists of a one-dimensional transverse-field Ising model, as discussed in the first example above, with each spin coupled to a heat bath of harmonic oscillators. The Hamiltonian reads... 

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... 

In the context of our interest in quantum phase transitions, however, we note that the accuracy of the DMRG method suffers greatly in the vicinity of quantum critical points. This was shown explicitly in two studies of the onedimensional Ising model in a transverse field, as given by the Hamiltonian of Eq. Legaza and Fath studied chains of up to 300 sites and found... 

It is not difficult to show that the special case of two singlets maps to the problem of an Ising model in a transversal field. The hamiltonian is then written as... 

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