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arXiv:2402.02794

Convolutional restricted Boltzmann machine (CRBM) correlated variational wave function for the Hubbard model on a square lattice: Mott metal-insulator transition

V., Karthik, Amal Medhi

Published Feb 5, 2024• 1 citation
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Abstract

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 $t'\neq 0$. 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 $U_c$, 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.

DOI
10.48550/arxiv.2402.02794

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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 UUU. We consider two cases for the next-nearest-neighbor hopping parameter, e.g., t′=0t'=0t′=0 as well as a frustrated model case with t′≠0t'\neq 0t′=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 UUU is increased, the system undergoes a first-order Mott transition to an insulating state at a critical UcU_cUc​, the value of which depends upon t′t'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′t't′, we find some interesting qualitative differences in the way the Mott transition takes place in the two cases.
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 with 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 UUU. We consider two cases for the next-nearest-neighbor hopping parameter, e.g., ${t}^{\ensuremath{'}}=0$ as well as a frustrated model case with ${t}^{\ensuremath{'}}\ensuremath{\ne}0$. 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 UUU is increased, the system undergoes a first-order Mott transition to an insulating state at a critical Uc{U}_{c}Uc​, the value of which depends on ${t}^{\ensuremath{'}}$. The Mott state in both cases is spin gapped with long-range antiferromagnetic (AFM) order. Remarkably, the AFM 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 ${t}^{\ensuremath{'}}$, we find some interesting qualitative differences in the way the Mott transition takes place in the two cases.
We consider a restricted Boltzmann Machine (RBM) correlated BCS wave function as the ground state of the two-dimensional Hubbard model and study its electronic and magnetic properties as a function of hole doping. We compare the results with those obtained by using conventional Jastrow projectors. The results show that the RBM wave function outperforms the Jastrow projected ones in the underdoped region inmterms of the variational energy. Computation of superconducting (SC) correlations in the model shows that the RBM wave function gives slightly weaker SC correlations as compared to the Jastrow projected wave functions. A significant advantage of the RBM wave function is that it spontaneously gives rise to strong antiferromagnetic (AF) correlations in the underdoped region even though the wave function does not incorporate any explicit AF order. In comparison, AF correlations in the Jastrow projected wave functions are found to be very weak. These and other results obtained show that the RBM wave function provides an improved description of the phase diagram of the model. The work also demonstrates the power of neural-network quantum state (NQS) wave functions in the study of strongly correlated electron systems.
We investigate the phase diagram of the square lattice bilayer Hubbard model at half-filling with the variational Monte Carlo method for both the magnetic and the paramagnetic case as a function of the interlayer hopping and on-site Coulomb repulsion U . With this study we resolve some discrepancies in previous calculations based on the dynamical mean-field theory, and we are able to determine the nature of the phase transitions between metal, Mott insulator and band insulator. In the magnetic case we find only two phases: an antiferromagnetic Mott insulator at small for any value of U and a band insulator at large . At large U values we approach the Heisenberg limit. The paramagnetic phase diagram shows at small a metal to Mott insulator transition at moderate U values and a Mott to band insulator transition at larger U values. We also observe a re-entrant Mott insulator to metal transition and metal to band insulator transition for increasing in the range of . Finally, we discuss the phase diagrams obtained in relation to findings from previous studies based on different many-body approaches.
Abstract. We investigate the phase diagram of the square lattice bilayer Hubbard model at half filling with the variational Monte Carlo method for both the magnetic and the paramagnetic case as a function of the interlayer hopping t ⊥ and on-site Coulomb repulsion U. With this study we resolve some discrepancies in previous calculations based on the dynamical mean field theory, and we are able to determine the nature of the phase transitions between metal, Mott insulator and band insulator. In the magnetic case we find only two phases: An antiferromagnetic Mott insulator at small t ⊥ for any value of U and a band insulator at large t⊥. At large U values we approach the Heisenberg limit. The paramagnetic phase diagram shows at small t ⊥ a metal to Mott insulator transition at moderate U values and a Mott to band insulator transition at larger U values. We also observe a re-entrant Mott insulator to metal transition and metal to band insulator transition for increasing t ⊥ in the range of 5.5t < U < 7.5t. Finally, we discuss the obtained phase diagrams in relation to previous studies based on different many-body approaches. PACS numbers: 71.10.Fd, 71.27.+a, 71.30.+h 1.
We investigate the Mott transition in a two-dimensional two-orbital Hubbard model on a square lattice at half filling by using the variational Monte Carlo method. As a variational trial wave function, we consider Gutzwiller's on-site correlations and nearest-neighbor doublon-holon correlations. We calculate the ground-state energy and the other physical quantities, all of which clearly signal a first-order Mott transition. We thus obtain the phase diagram at zero temperature, where we find the coexistence region of the metal and the Mott insulator. We discuss how Hund's coupling affects the nature of the Mott transition in this system.
We consider the one-band Hubbard model on the square lattice by using variational and Green's function Monte Carlo methods, where the variational states contain Jastrow and backflow correlations on top of an uncorrelated wave function that includes BCS pairing and magnetic order. At half-filling, where the ground state is antiferromagnetically ordered for any value of the on-site interaction UUU, we can identify a hidden critical point UMott{U}_{\mathrm{Mott}}UMott​, above which a finite BCS pairing is stabilized in the wave function. The existence of this point is reminiscent of the Mott transition in the paramagnetic sector and determines a separation between a Slater insulator (at small values of UUU), where magnetism induces a potential energy gain, and a Mott insulator (at large values of UUU), where magnetic correlations drive a kinetic energy gain. Most importantly, the existence of UMott{U}_{\mathrm{Mott}}UMott​ has crucial consequences when doping the system: We observe a tendency for phase separation into hole-rich and hole-poor regions only when doping the Slater insulator, while the system is uniform by doping the Mott insulator. Superconducting correlations are clearly observed above UMott{U}_{\mathrm{Mott}}UMott​, leading to the characteristic dome structure in doping. Furthermore, we show that the energy gain due to the presence of a finite BCS pairing above UMott{U}_{\mathrm{Mott}}UMott​ shifts from the potential to the kinetic sector by increasing the value of the Coulomb repulsion.