Direct detection of metal-insulator phase transitions using modified Backus-Gilbert method
- Dr. Maksim ULYBYSHEV
- Dr. Maksim ULYBYSHEV (University of Regensburg)
The detection of the (semi)metal-insulator phase transition can be extremely difficult if the local order parameter which characterizes the ordered phase is unknown. In some cases, it is even impossible to define a local order parameter: the most prominent example of such system is the spin liquid state. This state was proposed to exist in the Hubbard model on the hexagonal lattice in a region between the semimetallic phase and the antiferromagnetic insulator phase. The existence of this phase has been the subject of long debates.
Another example with similar difficulties is graphene in magnetic field: it is definitely in the insulating state, which is confirmed by experiments, but the type of ordering is unknown. There are several types of local order that have been proposed so far. Some of them (for instance, the Charge Density Wave) are even impossible to detect in a usual way where the small mass term is added as a seed for the phase transition and the corresponding susceptibility is calculated in the limit of zero mass. The problem with this approach appears because the mass term for the Charge Density wave produces a sign problem.
Thus, some alternative methods for the detection of the phase transitions in such cases should be developed. We modified the Backus-Gilbert method of analytical continuation which was previously used in the calculation of the pion quasiparticle mass in lattice QCD. The modification of the method consists of the introduction of the Tikhonov regularization scheme which was used to treat the ill-conditioned kernel. Furthermore, in some cases, additional averaging over judiciously chosen intervals in Euclidean time was introduced in order to suppress statistical fluctuations in the input data.
This modified Backus-Gilbert method is applied to the Euclidean propagators in momentum space calculated using the Hybrid Monte Carlo algorithm. In this way, it's possible to reconstruct the full dispersion relation and to estimate the mass gap, which is the direct signal of the transition to the insulating state. We demonstrate the ability of the method in the calculations for the Hubbard model on the hexagonal lattice and for graphene in a magnetic field. We also apply the method to the metal-insulator phase transition in the Hubbard and Hubbard-Coulomb models on the square lattice. The access to the full dispersion relation gives us a possibility to check the appearance of the momentum-dependent mass gap ("pseudogap" state) and it's dependence on the details of electron-electron interaction.
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Applications Beyond QCD