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arXiv:1409.5161 · Physical Review B

Entanglement properties of the antiferromagnetic-singlet transition in the Hubbard model on bilayer square lattices

Chia‐Chen Chang, Rajiv R. P. Singh, Richard T. Scalettar

Published Oct 10, 2014• 13 citations
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Abstract

We calculate the bipartite R\'enyi entanglement entropy of an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}2$ bilayer Hubbard model using a determinantal quantum Monte Carlo method recently proposed by Grover [Phys. Rev. Lett. 111, 130402 (2013)]. Two types of bipartition are studied: (i) one that divides the lattice into two $L\ifmmode\times\else\texttimes\fi{}L$ planes and (ii) one that divides the lattice into two equal-size ($L\ifmmode\times\else\texttimes\fi{}L/2\ifmmode\times\else\texttimes\fi{}2$) bilayers. We compare our calculations with those for the tight-binding model studied by the correlation matrix method. As expected, the entropy for bipartition (i) scales as ${L}^{2}$, while the latter scales with $L$ with possible logarithmic corrections. The onset of the antiferromagnet to singlet transition shows up by a saturation of the former to a maximal value and the latter to a small value in the singlet phase. We comment on the large uncertainties in the numerical results with increasing $U$, which would have to be overcome before the critical behavior and logarithmic corrections could be quantified.

DOI
10.1103/physrevb.90.155113

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We calculate the bipartite R'enyi entanglement entropy of an L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fi2L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}2L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fi2 bilayer Hubbard model using a determinantal quantum Monte Carlo method recently proposed by Grover [Phys. Rev. Lett. 111, 130402 (2013)]. Two types of bipartition are studied: (i) one that divides the lattice into two L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL planes and (ii) one that divides the lattice into two equal-size (L\ifmmode×\else\texttimes\fiL/2\ifmmode×\else\texttimes\fi2L\ifmmode\times\else\texttimes\fi{}L/2\ifmmode\times\else\texttimes\fi{}2L\ifmmode×\else\texttimes\fiL/2\ifmmode×\else\texttimes\fi2) bilayers. We compare our calculations with those for the tight-binding model studied by the correlation matrix method. As expected, the entropy for bipartition (i) scales as L2{L}^{2}L2, while the latter scales with LLL with possible logarithmic corrections. The onset of the antiferromagnet to singlet transition shows up by a saturation of the former to a maximal value and the latter to a small value in the singlet phase. We comment on the large uncertainties in the numerical results with increasing UUU, which would have to be overcome before the critical behavior and logarithmic corrections could be quantified.
We examine the entanglement properties of the spin-half Heisenberg model on the two-dimensional square-lattice bilayer based on quantum Monte Carlo calculations of the second R\'enyi entanglement entropy. In particular, we extract the dominant area-law contribution to the bipartite entanglement entropy that shows a nonmonotonous behavior upon increasing the interlayer exchange interaction: a local maximum in the area-law coefficient is located at the quantum critical point separating the antiferromagnetically ordered region from the disordered dimer-singlet regime. Furthermore, we consider subleading logarithmic corrections to the R\'enyi entanglement entropy scaling. Employing different subregion shapes, we isolate the logarithmic corner term from the logarithmic contribution due to Goldstone modes that is found to be enhanced in the limit of decoupled layers. At the quantum critical point, we estimate a contribution of 0.016(1)0.016(1)0.016(1) due to each ${90}^{\ensuremath{\circ}}$ corner. This corner term at the SU(2) quantum critical point deviates from the Gaussian theory value, while it compares well with recent numerical linked cluster calculations on the bilayer model.
We analyze the critical properties and the entanglement scaling at the quantum critical points of the spin-half XY model on a two-dimensional square-lattice bilayer and necklace lattice, based on quantum Monte Carlo simulations on finite tori and for different subregion shapes. For both models, the finite-size scaling of the transverse staggered spin structure factor is found in accord with a quantum critical point described by the two-component, three-dimensional ${\ensuremath{\phi}}^{4}$ theory. The second R\'enyi entanglement entropy in the absence of corners along the subsystem boundary exhibits area-law scaling in both models, with an area-law prefactor of 0.0674(7)0.0674(7)0.0674(7) [0.0664(4)][0.0664(4)][0.0664(4)] for the bilayer (necklace) model, respectively. Furthermore, the presence of ${90}^{\ensuremath{\circ}}$ corners leads to an additive logarithmic term in both models. We estimate a contribution of $\ensuremath{-}0.010(2)$ $[\ensuremath{-}0.009(2)]$ due to each ${90}^{\ensuremath{\circ}}$ corner to the logarithmic correction for the bilayer (necklace) model, and compare our findings to recent numerical linked-cluster calculations and series expansion results on related models.
We compute the von Neumann and generalized R\'enyi entanglement entropies in the ground-state of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square lattice using the modified spin-wave theory for finite lattices. The addition of a staggered magnetic field to regularize the zero modes associated with symmetry breaking is shown to be essential for obtaining well-behaved values for the entanglement entropy. The von Neumann and R\'enyi entropies obey an area law with additive logarithmic corrections, and are in good quantitative agreement with numerical results from valence bond quantum Monte Carlo and density matrix renormalization group calculations. We also compute the spin fluctuations and observe a multiplicative logarithmic correction to the area law in excellent agreement with quantum Monte Carlo calculations.
Employing a combination of functional renormalization group calculations and projective determinantal quantum Monte Carlo simulations, we examine the Hubbard model on the square lattice bilayer at half filling. From this combined analysis, we obtain a comprehensive account of the ground-state phase diagram with respect to the extent of the system's metallic and (antiferromagnetically ordered) Mott-insulating as well as band-insulating regions. By means of an unbiased functional renormalization group approach, we exhibit the antiferromagnetic Mott-insulating state as the relevant instability of the free metallic state, induced by any weak finite on-site repulsion. Upon performing a careful analysis of the quantum Monte Carlo data, we resolve the difficulty of identifying this antiferromagnetic ground state for finite interlayer hopping in the weak-coupling regime, where nonmonotonic finite-size corrections are shown to relate to the two-sheeted Fermi surface structure of the metallic phase. On the other hand, quantum Monte Carlo simulations are well suited to identify the transition between the Mott-insulating phase and the band insulator in the intermediate-to-strong coupling regime. Here, we compare our numerical findings to indications for the transition region obtained from the functional renormalization group procedure.
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.
We perform the finite-temperature determinant quantum Monte Carlo simulation for the attractive Hubbard model on the half-filled bilayer square lattice. Recent progress on optical lattice experiments lead us to investigate various single-particle properties such as momentum distribution and double occupancies which should be easily measured in cold-atom experiments. The pair-pair and the density-density correlations have been studied in detail and, through finite-size scaling, we show that there is no competing charge density wave order in the bilayer band-insulator model and that the superfluid phase is the stable phase for the interaction range ∣U∣/t=5−−10|U|/t=5--10∣U∣/t=5−−10. We show the existence of two energy scales in the system as we increase the attractive interaction, one governing the phase coherence and the other one corresponding to the molecule formation. In the end, we map out the full $T\ensuremath{-}U$ phase diagram and compare the Tc{T}_{c}Tc​ obtained through the mean-field analysis. We observe that the maximum Tc/t(=0.27){T}_{c}/t(=0.27)Tc​/t(=0.27) occurs for ∣U∣/t=6|U|/t=6∣U∣/t=6, which is roughly twice the reported Tc{T}_{c}Tc​ of the single-layer attractive Hubbard model.