I derive a mapping of the Hubbard and t-J models onto a description in terms of spins and spin-less fermions. This transformation can be combined with the Fedotov-Popov fermionisation technique, where spin-1/2 systems are mapped onto two-component lattice fermions with imaginary chemical potential. The result is a generalisation of the Fedotov-Popov formulation to the case of finite doping that makes it possible to treat the Hubbard and t-J models as spin-models in the presence of a finite density of holes and doublons. While strongly interacting lattice fermions near half-filling is an extremely challenging problem, spin-systems can be efficiently treated with current simulation techniques.
I derive a dual description of lattice fermions, specifically focusing on the t-J and Hubbard models, that allow diagrammatic techniques to be employed efficiently in the strongly correlated regime, as well as for systems with a restricted Hilbert space. These constructions are based on spin-charge transformation, where the lattice fermions of the original model are mapped onto spins and spin-less fermions. This mapping can then be combined with Popov-Fedotov fermionisation, where the spins are mapped onto lattice fermions with imaginary chemical potential. The resulting models do not contain any large expansion parameters, even for strongly correlated systems. Also, they exhibit dramatically smaller corrections to the density matrix from nonlinear terms in the Hamiltonian. The combination of these two properties means that they can be addressed with diagrammatic methods, including simulation techniques based on stochastic sampling of diagrammatic expansions.
We find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential
μ=iθ. We show, by using a large
N-expansion, that the partition functions of the two models are related by a simple shift
θ→θ+π. This condition maps the BCS--BEC crossover of attractive fermions to a Bose--Fermi crossover (fermion-like occupation) of repulsive bosons. Central feature of this correspondence plays the thermal kernel
g(βE,ϕ), whose analytic continuation
gB(βE,ϕ)=gF(βE,ϕ+π) governs the bosonic and fermionic sectors. Interestingly, we are able to find that the special angles
ϕ=2π/3,4π/3 for fermions correspond to
ϕ=π/3,5π/3 for bosons, marking the boundaries of a universal thermal window. We further argue that the present mechanism shows that fermionization can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. This suggests that it is a new way of fermionization (not a change in statistics but a fermion-like behaviour) unlike the Tonks--Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Essentially, the phase
ϕ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behaviour without hard-core constraints or infinite repulsion but only by using thermodynamics. We derive the gap equation and number equation for the bosonic model, highlighting the role of the imaginary chemical potential as a statistical regulator. Our results provide a unified framework for understanding crossovers in interacting lattice systems.
We present diagrammatic approximations to the spin dynamics of the 2d Heisenberg antiferromagnet for all temperatures, employing an auxiliary-fermion representation. The projection onto the physical subspace is effected by introducing an imaginary-valued chemical potential as proposed by Popov and Fedotov. The method requires that the fermion number at any lattice site is strictly conserved. We compare results obtained within a self-consistent approximation using two different auxiliary-particle projection schemes, (1) exact and (2) on average. Significant differences between the two are found at higher temperatures, whereas in the limit of zero temperature (approaching the magnetically ordered ground state) identical results emerge from (1) and (2), providing the qualitatively correct dynamical scaling behavior. An interpretation of these findings is given. We also present in some detail the derivation of the approximation, which goes far beyond mean-field theory and is formulated in terms of complex-valued spectral functions of auxiliary fermions.
Using a dual representation of lattice fermion models that is based on spin-charge transformation and fermionization of the original description, I derive an algorithm for diagrammatic Monte Carlo simulation of strongly correlated systems. This scheme allows eliminating large expansion parameters, as well as large corrections to the density matrix that generally prevent diagrammatic methods from being efficient in this regime. As an example, I compute the filling factor for the Hubbard model at infinite on-site repulsion and compare the results to controllable data obtained from numerical linked-cluster expansion. I find excellent agreement between the two methods, as well as rapid convergence of the diagrammatic series. I also report results for the momentum distribution and kinetic energy of the electrons.