Phase diagram of MI/SF transition Bose-Hubbard model

[HungryBunny]

Hi all,

I am currently doing an undergraduate research project on the Bose-Hubbard model with hardcore bosons, and I am studying it theoretically using the Monte Carlo Stochastic Series Expansion method. I would very much like to ask my supervisor this question, but he seems extremely busy lately so I will try asking here first.

What order parameter does one use to determine if for a given set of Hamiltonian parameter values, the system is in a Mott Insulator or superfluid state? I've read many papers and some say to calculate compressibility, others say to calculate superfluid density. As for superfluid density, how does one tell exactly from the value that the system is in MI/SF state? Papers I've read say MI state has a low superfluid density, while SF state has a high superfluid density. What does it mean by low or high? It seems so subjective.

 

[eljose79]

Thank you for your question regarding the Bose-Hubbard model with hardcore bosons and the determination of the Mott insulator (MI) or superfluid (SF) state using the Monte Carlo Stochastic Series Expansion method. This is an interesting and important topic in the field of condensed matter physics.

Firstly, to determine the MI/SF state, one can use either the compressibility or the superfluid density as an order parameter. Both are valid methods and have been used in various research papers. The choice of which one to use may depend on the specific system and the research goals.

The compressibility is a measure of the system's response to changes in chemical potential, and it can be calculated using the fluctuation-dissipation theorem. In the MI state, the compressibility is expected to be low due to the presence of a gap in the excitation spectrum. On the other hand, in the SF state, the compressibility is expected to be high due to the presence of gapless excitations. Therefore, a low compressibility can indicate the presence of a MI state, while a high compressibility can indicate the presence of a SF state.

The superfluid density, on the other hand, is a measure of the system's response to changes in phase. It can be calculated using the winding number or the current-current correlation function. In the MI state, the superfluid density is expected to be low due to the lack of long-range phase coherence. In contrast, in the SF state, the superfluid density is expected to be high due to the presence of long-range phase coherence. Therefore, a low superfluid density can indicate the presence of a MI state, while a high superfluid density can indicate the presence of a SF state.

In terms of determining the exact value that indicates the MI/SF state, this may vary depending on the specific system and the research goals. In general, a clear distinction between a low and high value can be made by comparing with the expected values for the MI and SF states. This can also be confirmed by studying the behavior of the order parameter as a function of the system's parameters (such as the interaction strength and lattice potential). In some cases, a phase diagram can also be constructed to show the regions of MI and SF states for different parameter values.

I hope this helps to clarify your question. If you have any further questions, please do not hesitate to ask. Best of luck with your research