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Computing fidelity at magnetic quantum phase transitions

Quantum phase transitions (QPT) play an important role in condensed matter systems. Traditional investigations of those QPT usually need the definition of an order parameter, which itself already requires a deeper understanding of the physical system. When a system undergoes a quantum phase transition, the ground-state wave-function shows a change of nature, which can be monitored using the fidelity concept.

Recently it was pointed out that quantum phase transitions can also be detected by using a quantity called fidelity. Being rooted in quantum information theory this approach stayed relatively unregarded in the condensed matter community. Reasons might be found in the fact that there is only a small class of analytically solvable problems, but also that the used numerical techniques such as Exact Diagonalization (ED) and Density Matrix Group Renormalization (DMRG) are basically restricted to low dimension.

However, one of the most powerful tools, the Quantum Monte Carlo (QMC) technique, up to now has not been considered to calculate fidelity. In our paper, we introduce two QMC schemes that allow the computation of fidelity and its susceptibility for large interacting many-body systems.

On one hand, it turns out that the a recently proposed projector QMC at zero temperature allows for direct calculation of fidelity. This is however restricted a class of sign-problem free systems with a singlet ground state, such as the AF Heisenberg Model on a bipartite lattice. We show the efficiency of our scheme on the 1/5-depleted square lattice, where fidelity estimators show marked behaviours at two successive quantum phase transitions.

On the other hand, the leading term of fidelity defines a susceptibility which can also be used to detect QPTs. We have shown how this quantity can be calculated within the even more general SSE formalism. This QMC scheme can be applied to an even larger class of sign-problem free models. In particular this method allows for the investigation of the finite size scaling of the fidelity susceptibility. We also develop a scaling theory which relates the divergence of the fidelity susceptibility to the critical exponent of the correlation length. A good agreement is found with the numerical results.

Reference : D. Schwandt, F. Alet, and S. Capponi, Phys. Rev. Lett. 103, 170501 (2009)