## The generalised t-V model - what is it?

The generalised t-V model is a model of interacting fermions, where the interactions are repulsive and of long range. The Hamiltonian of the model is:

where

*c*are the fermionic operators,*n*are the particle number operators,*L*is the system size,*t*is the kinetic energy, and*Um*is the potential energy between two fermions that are*m*sites apart.*p*is the maximum interaction range.Such a setup causes fermions to stay at least

*p*sites away from each other, otherwise a huge energy penalty is caused (*t<<Um*). We can then consider changing the density of the fermions in the system. This causes emergence of two phases (see Fig. 1):- Luttinger liquid phase - where elementary excitations are phonon-like density fluctuations - happens for almost all densities
*Q*, and - Mott insulating phase - where despite a presence of the charge carriers the system is insulating due to their repulsive interactions - present only for critical densities
*Qc*= 1/*k*.

## Solving the generalised t-V model

We are interested in solving the generalised t-V model by using already existing analytical tools, such as Gómez-Santos [1] method or the Dias method [2]. We have adapted the Dias method in order to expand the considered interaction range beyond

*p*= 1. We have also used strong coupling expansion - a method widely used in lattice field theory community - in order to achieve high-perturbation-order estimates of the system near the Mott insulating densities.We present a short comparison of the methods below in Table 1.

## Charge-density-wave phases in the model with unusual potentials

If one considers the potential energy not to be steadily decreasing, the model can exhibit non-trivial behaviour and the emergence of unusual charge-density-wave phases [3,4]. An example phase diagram is shown in Fig. 2 with corresponding phases described in Table 2.

## References

[1] Gómez-Santos, Phys. Rev. Lett. 70(24), 3780 (1993).

[2] Dias, Phys. Rev. B 62, 7791 (2000).

[3] P. Schmitteckert and R. Werner, Phys. Rev. B 69, 195115 (2004).

[4] T. Mishra, J. Carrasquilla, and M. Rigol, Phys. Rev. B 84, 115135 (2011).

[2] Dias, Phys. Rev. B 62, 7791 (2000).

[3] P. Schmitteckert and R. Werner, Phys. Rev. B 69, 195115 (2004).

[4] T. Mishra, J. Carrasquilla, and M. Rigol, Phys. Rev. B 84, 115135 (2011).