Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

Abstract

The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one-and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to hole doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak d-wave superconducting order in the ground state for the doping concentration less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.

Approximate expression for the ground-state energy of the two- and three-dimensional Hubbard model at arbitrary filling obtained from dimensional scaling

“…4 shows that linear approximation overestimates the per-site ground-state energy, while the nonlinear approximation underestimates it. The same trend is apparent in Table 2, which displays more data [25].…”

Period 4 stripe in the extended two-dimensional Hubbard model

“…Meanwhile, the two-dimensional (2D) Hubbard model [33] -which has been suggested as the most elementary microscopic model that may reproduce the essential features of the cuprates' phase diagram -has been a prominent subject of intense theoretical and numerical investigations. Much attention has been given especially on the underdoped region in the strongly correlated regime, where several low-energy states are very closely competing, including a uniform d-wave superconducting (SC) state [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] and various stripe states [20][21][22][23][52][53][54][55][56][57][58][59][60][61] with or without coexisting superconducting order. A similar competition can also be found for the t − J model -the effective model in the strong coupling limit [62][63][64][65][66][67][68][69][70][71].…”

Energy of fermionic ground states with low-entanglement single-reference expansions, and tensor-based strong-coupling extensions of the coupled-cluster method

“…On the other hand, as discussed in section IV B, at some large coupling strength, TCICC could become more compact than TCC, and thus better suited to even stronger couplings. There are also well-known strongly correlated wave functions that have a tensor network representation of the coefficients and are good variational ground states of strongly correlated systems 16,19,23 , suggesting that the direct parametrization of wave function coefficients in TCICC is well suited to that regime. Finally, although TCICC can also be seen as a low-entanglement version of the CI method, since the equations (36), (38) and (40) involve the CI coefficients, it is in fact very different from CI: First, Hilbert space is not truncated in the number of particle-hole excitations, then, the coefficients are not computed variationally or by matrix diagonalization, but instead the tensors defining the coefficients are obtained by solving non-linear equations, and finally, the result is size-extensive because the energy is linked.…”

“…To derive the equations, we have used the excited particle and hole operators, Eqs. (23), and the representation (33) of the Hamiltonian. However, although that representation is convenient for that task, it remains a quite lengthy derivation.…”