Transverse-Field Ising Chain

The quantum-to-classical mapping in the two examples above resulted in systems where space and time directions appear symmetrically, implying a dynamical exponent z = 1 from the outset. We now turn to an example where space and time directions scale differently, and where the dynamical exponent has to be determined from the simulation data. 

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 

Here and are the creation and destruction operators of harmonic oscillator k coupled to spin i. The oscillator frequencies and coupling constants Ck are chosen such that the spectral function /(co) = 4k = 

The quantum-to-classical mapping for this system follows the same Feynman path integral procedure used in the first example. The harmonic oscillator degrees of freedom lead to Gaussian integrals and can thus be integrated out exactly. The resulting classical Hamiltonian reads 

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D. S. Fisher, Phys. Rev. Lett., 69, 534 (1992). Random Transverse-Field Ising Spin Chains. 

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... 

P. Werner, K. Volker, M. Troyer, and S. Chakravarty (2005) Phase diagram and critical exponents of a dissipative Ising spin chain in a transverse magnetic field. Phys. Rev. Lett. 94, p. 047201... 

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