Bosons in Two Dimensions

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. 

The minimal model for describing the superconductor-insulator transition in the general case of both charge and phase fluctuations being relevant is the boson Hubbard model with a random local chemical potential.The Hamiltonian (defined on a square lattice) takes the form 

U is the onsite repulsion, p is the chemical potential, z is the number of nearest neighbors, and Vi represents the random onsite potential. The hopping strength is given by t, and 4 , 4 / are the boson creation and destruction operators at site L The number operator is given by N/ = 

If the boson density is an integer (per site), and, in the absence of disorder, charge (amplitude) fluctuations are small. If we set = ( and integrate out the amplitude fluctuations, we obtain a phase-only model that can be written as an 0(2) quantum rotor model  

This Hamiltonian describes, among other systems, an array of coupled Joseph- 

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G. Schmid, S. Todo, M. Troyer, and A. Dorneich (2002) Finite-temperature phase diagram of hard-core bosons in two dimensions. Phys. Rev. Lett. 88, p. 167208... 

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