A planar model for interacting fermions

Trong Tuong Truong, K. D. Schotte

We show that a soluble Hamiltonian for interacting fermions in one-dimensional conductors generates a classical double Coulomb gas of "electric" and "magnetic" charges and thereby establishes its equivalence with the generalized Villain model in the presence of a fourfold symmetry-breaking term recently studied by L. P. Kadanoff. We use the exact solution of the fermion model to demonstrate how the eight-vertex structure of the mutual fermion scatterings controls its critical behavior.

Vincent Rivasseau

Ning Bao, Patrick M. Hayden, Grant Salton +1 more

The Hubbard model may be the simplest model of particles interacting on a lattice, but simulation of its dynamics remains beyond the reach of current numerical methods. In this article, we show that general quantum computations can be encoded into the physics of wave packets propagating through a planar graph, with scattering interactions governed by the fermionic Hubbard model. Therefore, simulating the model on planar graphs is as hard as simulating quantum computation. We give two different arguments, demonstrating that the simulation is difficult both for wave packets prepared as excitations of the fermionic vacuum, and for hole wave packets at filling fraction one-half in the limit of strong coupling. In the latter case, which is described by the t - J model, there is only reflection and no transmission in the scattering events, as would be the case for classical hard spheres. In that sense, the construction provides a quantum mechanical analog of the Fredkin–Toffoli billiard ball computer.

Seth Grable, Noah Kamm, Harsh Mathur

Tao Yang, Bangming Wang, Peng Song +2 more

Quantum phase transitions in quantum magnets constitute an important field that has attracted great interests. Although a variety of analytic and numerical methods have been introduced in this direction, faithful fermionic descriptions are still desired because they can transform the spin models to a form tractable by conventional many-body techniques, yielding more transparent physical pictures. We generalize the Chern-Simons fermionization approach and apply it to the XXZ quantum magnets. After fermionzation, the Ising interaction of the XXZ spin model leads to a fermion-fermion interaction. We show that the additional fermion-fermion interaction generates an interesting phase transition between two fermionic ordered states, i.e., the Chern-Simons superconductor and the Chern-Simons exciton insulator. We also demonstrate that this transition in the fermionic language essentially describes the quantum phase transition between the planar and out-of-plane N\'eel orders in the spin picture. The fermionic mean-field theory further leads to a nonlinear $\ensuremath{\sigma}$ model that describes the quantum phase transition, which is further supported by our density matrix renormalization group calculations in the original spin model. Our work introduces a fermionic interpretation of quantum phase transitions and advances the existing knowledge of quantum magnets.