We introduce an efficient way to improve the accuracy of projected wave functions, widely used to study the two-dimensional Hubbard model. Taking the clue from the backflow contribution, whose relevance has been emphasized for various interacting systems on the continuum, we consider many-body correlations to construct a suitable approximation for the ground state at intermediate and strong couplings. In particular, we study the phase diagram of the frustrated $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model on the square lattice and show that, thanks to backflow correlations, an insulating and nonmagnetic phase can be stabilized at strong coupling and sufficiently large frustrating ratio ${t}^{\ensuremath{'}}/t$.
We show that backflow correlations in the variational wave function for the Hubbard model greatly improve the previous results given by the Slater-Jastrow state, usually considered in this context. We provide evidence that, within this approach, it is possible to have a satisfactory connection with the strong-coupling regime. Moreover, we show that, for the Hubbard model on the lattice, backflow correlations are essentially short range, inducing an effective attraction between empty (holons) and doubly occupied sites (doublons). In the presence of frustration, we report the evidence that the metal to Mott-insulator transition is marked by a discontinuity of the double occupancy, together with a similar discontinuity of the kinetic term that does not change the number of holons and doublons, while the other kinetic terms are continuous across the transition. Finally, we show the estimation of the charge gap, obtained by particle-hole excitations \`a la Feynman over the ground-state wave function.
Unique features of the nonmagnetic insulator phase are revealed, and the phase diagram of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model containing the diagonal transfers ${t}^{\ensuremath{'}}$ on a square lattice is presented. Using the path-integral renormalization group method, we find an antiferromagnetic phase for small next-nearest-neighbor transfer ${t}^{\ensuremath{'}}$ and a stripe (or collinear) phase for large ${t}^{\ensuremath{'}}$ in the Mott insulating region of the strong on-site interaction
U. For intermediate ${t}^{\ensuremath{'}}∕t\ensuremath{\sim}0.7$ at large
U∕t>7, we find a longer-period antiferromagnetic-insulator phase with $2\ifmmode\times\else\texttimes\fi{}4$ structure. In the Mott insulating region, we also find a quantum spin liquid (in other words, a nonmagnetic insulator) phase near the Mott transition to paramagnetic metals for the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model on the square lattice as well as on the anisotropic triangular lattice. Correlated electrons often crystallize to the Mott insulator usually with some magnetic orders, whereas the ``quantum spin liquid" has been a long-sought issue. We report numerical evidence that a nonmagnetic insulating phase gets stabilized near the Mott transition with remarkable properties: The two-dimensional Mott insulators on geometrically frustrated lattices contain a phase with gapless spin excitations and degeneracy of the ground state in the whole Brillouin zone of the total momentum. The obtained vanishing spin renormalization factor suggests that spin excitations do not propagate coherently in contrast to conventional phases, where there exist either magnons in symmetry-broken phases or particle-hole excitations in paramagnetic metals. It imposes a constraint on the possible pictures of quantum spin liquids and supports an interpretation for the existence of an unconventional quantum liquid. The present concept is useful in analyzing a variety of experimental results in frustrated magnets including organic BEDT-TTF compounds and
3He atoms adsorbed on graphite.
We investigate the properties of the frustrated underdoped Hubbard model on the square lattice using two complementary approaches, i.e., the dynamical cluster extension of the dynamical mean-field theory and the variational Monte Carlo simulations of Gutzwiller-Jastrow wave functions with backflow corrections. We compare and discuss data for the energy and the double occupancies, as obtained from both approaches. At small dopings, we observe a rapid crossover from a weakly correlated metal at low interaction strength
U to a non-Fermi-liquid correlated state with strong local spin correlations. Furthermore, we investigate the stability of the correlated state against phase separation. We observe phase separation only for large values of
U or very large frustration. No phase separation is present for the parameter range relevant for the cuprates.
We study the interplay between electron correlation and disorder in the two-dimensional Hubbard model at half filling by means of a variational wave function that can interpolate between Anderson and Mott insulators. We give a detailed description of our improved variational state and explain how the physics of the Anderson-Mott transition can be inferred from equal-time correlations functions, which can be easily computed within the variational Monte Carlo scheme. The ground-state phase diagram is worked out in both the paramagnetic and the magnetic sector. Whereas in the former a direct second-order Anderson-Mott transition is obtained, when magnetism is allowed variationally, we find evidence for the formation of local magnetic moments that order before the Mott transition. Although the localization length increases before the Mott transition, we have no evidence for the stabilization of a true metallic phase. The effect of a frustrating next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ is also studied in some detail. In particular, we show that ${t}^{\ensuremath{'}}$ has two primary effects. The first one is the narrowing of the stability region of the magnetic Anderson insulator, also leading to a first-order magnetic transition. The second and most important effect of a frustrating hopping term is the development of a ``glassy'' phase at strong couplings, where many paramagnetic states, with disordered local moments, may be stabilized.
We use a convolutional restricted Boltzmann machine (CRBM) neural network to construct a variational wave function (WF) for the Hubbard model on a square lattice and study it using the variational Monte Carlo (VMC) method. In the wave function, the CRBM acts as a correlation factor to a mean-field BCS state. The number of variational parameters in the WF does not grow automatically with the lattice size and it is computationally much more efficient compared to other neural network based WFs. We find that in the intermediate to strong coupling regime of the model at half-filling, the wave function outperforms even the highly accurate long range backflow-Jastrow correlated wave function. Using the WF, we study the ground state of the half-filled model as a function of onsite Coulomb repulsion
U. We consider two cases for the next-nearest-neighbor hopping parameter, e.g.,
t′=0 as well as a frustrated model case with
t′=0. By examining several quantities, e.g., double occupancy, charge gap, momentum distribution, and spin-spin correlations, we find that the weekly correlated phase in both cases is paramagnetic metallic (PM). As
U is increased, the system undergoes a first-order Mott transition to an insulating state at a critical
Uc, the value of which depends upon
t′. The Mott state in both cases is spin gapped with long range antiferromagnetic (AF) order. Remarkably, the AF order emerges spontaneously from the wave function which does not have any explicitly broken symmetry in it. Apart from some quantitative differences in the results for the two values of
t′, we find some interesting qualitative differences in the way the Mott transition takes place in the two cases.