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We reconsider the Mott transition in the context of a two-dimensional fermion model with density-density coupling. We exhibit a Hilbert space mapping between the original model and the Double Lattice Chern-Simons theory at the critical point by use of the representation theory of the q-oscillator and Weyl algebras. The transition is further characterized by the ground state modification. The explicit mapping provides a new tool to further probe and test the detailed physical properties of the fermionic lattice model considered here and to enhance our understanding of the Mott transition(s).

The physical properties of strongly correlated electron systems are difficult to predict or even to describe, mainly because of the lack of suitable reliable tools to study them. Among these systems, the Mott Insulators (i.e., electronic systems which undergo a metal-insulator transition driven by correlations) have a prominent place. Most of the studies of the Mott transition are based on the microscopic dynamics of the electron system. The models are defined by an electron Hamiltonian that is then solved either by some approximation or by numerical methods. Both methods have their limitations, which have been discussed, e.g., in [

For the case of the Mott transition, we have applied the EFT method to a fermion model on the lattice with density-density coupling in a previous paper [

We start by considering the following Hamiltonian model:

where

This model can be mapped into the two-dimensional anisotropic Heisenberg (

where

The partition function of the two-dimensional Heisenberg model, in the Hamiltonian framework, can be written as:

where

where

where, we have made some abuse of notation since now

As it is known, the

where the

which codifies the zero-curvature condition of a “quantum scattering problem”. Here

where

where q is the deformation parameter. The corresponding L operators are:

where we have used

This means that the original model is mapped onto the q-oscillator model. It then becomes possible to study the states of the lattice fermion model (1) by analyzing the representations of the q-oscillator algebra.

For

for

To achieve a deeper understanding of the solution that we have just discussed, we may use a crucial property of the ZTE. Namely, the ZTE can be projected (or reduced) onto the Yang-Baxter equation after tracing out over one (temporal or spatial) direction. Tracing out over the y-column we obtain a one-dimensional Heisenberg

In order to identify the critical point with the values of the parameters in the fermion model (1) let us remind that the reduced one-dimensional model (which has a long history) have been solved in [

We will now study the representations of the q-Oscillattor algebra at the Mott transition point

which have cyclic representations for

Now we claim that the “corresponding” field theory at the level of the Hilbert space is a Double-Lattice-Chern Simons theory with abelian gauge group. To show this, first we shall impose periodic boundary conditions in the original fermion model, and compactify the manifold onto a torus such that the original square lattice matches with the lattice made by the homology cycles of this torus, and consider the Abelian C-S action [

this a topological gauge field theory with natural observables provided by Wilson Loops:

In holomorphic coordinates, the gauge field may be decomposed as:

where

The local Gauge transformations on the CS theory are defined by:

where

which satisfy a Weyl-algebra:

On the basis

the operators (29) act as:

where

Then, taken

Secondly, we note that the projection property of the ZTE implies that each row (or column) is a

Taking into account that the degrees of freedom of the Fermion Model (1) must be restricted to a square lattice, using the fact that the q-oscillator algebra splits into two Weyl algebras and using the parity of the original model, we deduce that the corresponding (equivalent) Field Theory at the Mott critical point of the fermion model (1) is a Double Lattice Chern-Simons theory:

with coupling constant

right), and where

the lattice-parameter) [

tion parameter

In this article, we have reconsidered the integrability of the two-dimensional density-density coupled fermion moldel (1), which follows from the solution of the Zamolodchikov’s Tetrahedron equation associated with the

G. R. Zemba is a member of CONICET (Argentina).