Construction of matrix product state for the Gutwiller projected variational wavefunctions

An accurate description of various quantum spin liquid states using tensor network methods remains notoriously challenging. For large quasi-1D systems, the density matrix renormalization group and related methods usually require significant computational resources and sometimes fail to converge to a satisfactory state. On the other hand, variational wavefunctions acquired from the Gutzwiller projection of gaussian fermionic theories has long served as both a theoretical starting point for construction of such spin liquid states and as an inspiration for numerical variational Monte Carlo (VMC) to calculate observables of interest. In this work, we examine a different method by exploring the possibility of constructing a matrix product state (MPS) representation for a Gutzwiller-projected state from two given MPS representations of gaussian fermionic theories. We investigate the complexity of different approaches to achieve this goal and test the methods on two copies of a single half-filled band of spin-1/2 fermionic spinons. We then apply this method to two MPS of multi-band fermionic spinon theories in an attempt to describe spin liquid states on a quasi-1D strips of triangular and kagome-like lattices and compare to the complexity of the traditional VMC approach.