# Mott Transition in the Hubbard Model on Anisotropic Honeycomb Lattice with Implications for Strained Graphene: Gutzwiller Variational Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

#### 2.1. The Anisotropic Hubbard Model

#### 2.2. Hartree–Fock Approximation

#### 2.3. Gutzwiller Wavefunction

#### 2.4. Gutzwiller Approximation and Its Variants

## 3. Results and Discussion

#### 3.1. Phase Diagram

#### 3.2. Effects of Strain on Measurable Quantities

## 4. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Coherent Potential Approximation

## Appendix B. Su–Schrieffer–Heeger Model for Graphene

## References and Notes

- Gutzwiller, M. Effect of Correlation on the Ferromagnetism of Transition Metals. Phys. Rev. Lett.
**1963**, 10, 159. [Google Scholar] [CrossRef] - Hubbard, J. Electron Correlations in Narrow Energy Bands. Proc. R. Soc. A
**1963**, 276, 238. [Google Scholar] [CrossRef] - Lieb, E.H.; Wu, F.Y. Absence of Mott transition in an exact solution of the short-range one-band model in one dimension. Phys. Rev. Lett.
**1968**, 20, 1445, Erratum in Phys. Rev. Lett.**1968**, 21, 192. [Google Scholar] [CrossRef] - Lieb, E.H.; Wu, F.Y. The one-dimensional Hubbard model: A reminiscence. Phys. A
**2003**, 321, 1. [Google Scholar] [CrossRef] [Green Version] - Hirsch, J.E. Two-dimensional Hubbard model: Numerical simulation study. Phys. Rev. B
**1985**, 31, 4403. [Google Scholar] [CrossRef] [Green Version] - Acquarone, M.; Ray, D.K.; Spałek, J. The Hubbard sub-band structure and the cohesive energy of narrow band systems. J. Phys. C Solid State Phys.
**1982**, 15, 959. [Google Scholar] [CrossRef] - Yokoyama, H.; Shiba, H. Variational Monte-Carlo Studies of Hubbard Model. II. J. Phys. Soc. Jpn.
**1987**, 56, 3582. [Google Scholar] [CrossRef] - Li, Y.M.; d’Ambrumenil, N. Sum rule and symmetry-controlled expansion for generalized Gutzwiller wave functions. Phys. Rev. B
**1992**, 46, 13928. [Google Scholar] [CrossRef] [PubMed] - Li, Y.M.; d’Ambrumenil, N. A new expansion for generalized Gutzwiller wave functions: Antiferromagnetic case. J. Appl. Phys.
**1993**, 73, 6537. [Google Scholar] [CrossRef] - Koch, E.; Gunnarsson, O.; Martin, R.M. Optimization of Gutzwiller wave functions in quantum Monte Carlo. Phys. Rev. B
**1999**, 59, 15632. [Google Scholar] [CrossRef] - Becca, F.; Sorella, S. Quantum Monte Carlo Approaches for Correlated Systems; For a Comprehensive Review of the Topic; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar] [CrossRef]
- Czarnik, P.; Rams, M.M.; Dziarmaga, J. Variational tensor network renormalization in imaginary time: Benchmark results in the Hubbard model at finite temperature. Phys. Rev. B
**2016**, 94, 235142. [Google Scholar] [CrossRef] [Green Version] - Schneider, M.; Ostmeyer, J.; Jansen, K.; Luu, T.; Urbach, C. The Hubbard model with fermionic tensor networks. arXiv
**2021**, arXiv:2110.15340. [Google Scholar] - Fishman, M.; White, S.R.; Stoudenmire, E.M. The ITensor Software Library for Tensor Network Calculations. SciPost Phys. Codebases
**2022**, 4. [Google Scholar] [CrossRef] - Martelo, L.M.; Dzierzawa, M.; Siffert, L.; Baeriswyl, D. Mott-Hubbard transition and antiferromagnetism on the honeycomb lattice. Z. Phys. B
**1997**, 103, 335. [Google Scholar] [CrossRef] - Le, D.A. Mott transition in the half-filled Hubbard model on the honeycomb lattice within coherent potential approximation. Mod. Phys. Lett. B
**2013**, 27, 1350046. [Google Scholar] [CrossRef] - Rowlands, D.A.; Zhang, Y.-Z. Disappearance of the Dirac cone in silicene due to the presence of an electric field. Chin. Phys. B
**2014**, 23, 037101. [Google Scholar] [CrossRef] [Green Version] - Sorella, S.; Otsuka, Y.; Yunoki, S. Absence of a Spin Liquid Phase in the Hubbard Model on the Honeycomb Lattice. Sci. Rep.
**2012**, 2, 992. [Google Scholar] [CrossRef] [Green Version] - Sorella, S.; Tosatti, E. Semi-Metal-Insulator Transition of the Hubbard Model in the Honeycomb Lattice. Eur. Lett.
**1992**, 19, 699. [Google Scholar] [CrossRef] - Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Katsnelson, M.I.; Grigorieva, I.V.; Dubonos, S.V.; Firsov, A.A. Two-dimensional gas of massless Dirac fermions in graphene. Nature
**2005**, 438, 197. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Tan, Y.-W.; Stormer, H.L.; Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature
**2005**, 438, 201. [Google Scholar] [CrossRef] - Katsnelson, M.I. The Physics of Graphene, 2nd ed.; Cambridge University Press: Cambridge, UK, 2020. [Google Scholar] [CrossRef]
- Schüler, M.; Rösner, M.; Wehling, T.O.; Lichtenstein, A.I.; Katsnelson, M.I. Optimal Hubbard models for materials with nonlocal Coulomb interactions: Graphene, silicene and benzene. Phys. Rev. Lett.
**2013**, 111, 036601. [Google Scholar] [CrossRef] [PubMed] - Tang, H.-K.; Laksono, E.; Rodrigues, J.N.B.; Sengupta, P.; Assaad, F.F.; Adam, S. Interaction-Driven Metal-Insulator Transition in Strained Graphene. Phys. Rev. Lett.
**2015**, 115, 186602. [Google Scholar] [CrossRef] [Green Version] - Zhang, L.; Ma, C.; Ma, T. Metal-Insulator Transition in Strained Graphene: A Quantum Monte Carlo Study. Phys. Status Solidi RRL
**2021**, 15, 2100287. [Google Scholar] [CrossRef] - Pasternak, M.P.; Nasu, S.; Wada, K.; Endo, S. High-pressure phase of magnetite. Phys. Rev. B
**1994**, 50, 6446. [Google Scholar] [CrossRef] - Goncharenko, I.N. Evidence for a Magnetic Collapse in the Epsilon Phase of Solid Oxygen. Phys. Rev. Lett.
**2005**, 94, 205701. [Google Scholar] [CrossRef] [PubMed] - Drozdov, A.P.; Eremets, M.I.; Troyan, I.A.; Ksenofontov, V.; Shylin, S.I. Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system. Nature
**2015**, 525, 73. [Google Scholar] [CrossRef] [Green Version] - Somayazulu, M.; Ahart, M.; Mishra, A.K.; Geballe, Z.M.; Baldini, M.; Meng, Y.; Struzhkin, V.V.; Hemley, R.J. Evidence for Superconductivity above 260 K in Lanthanum Superhydride at Megabar Pressures. Phys. Rev. Lett.
**2019**, 122, 027001. [Google Scholar] [CrossRef] [Green Version] - Celliers, P.M.; Millot, M.; Brygoo, S.; McWilliams, R.S.; Fratanduono, D.E.; Rygg, J.R.; Goncharov, A.F.; Loubeyre, P.; Eggert, J.H.; Peterson, J.L.; et al. Insul.-Met. Transit. Dense Fluid Deuterium. Science
**2018**, 361, 677. [Google Scholar] [CrossRef] [Green Version] - Feldner, H.; Meng, Z.Y.; Honecker, A.; Cabra, D.; Wessel, S.; Assaad, F.F. Magnetism of finite graphene samples: Mean-field theory compared with exact diagonalization and quantum Monte Carlo simulations. Phys. Rev. B
**2010**, 81, 115416. [Google Scholar] [CrossRef] - Potasz, P.; Güçlü, A.D.; Wójs, A.; Hawrylak, P. Electronic properties of gated triangular graphene quantum dots: Magnetism, correlations, and geometrical effects. Phys. Rev. B
**2012**, 85, 075431. [Google Scholar] [CrossRef] [Green Version] - Brito, F.M.O.; Li, L.; Lopes, J.M.V.P.; Castro, E.V. Edge magnetism in transition metal dichalcogenide nanoribbons: Mean field theory and determinant quantum Monte Carlo. Phys. Rev. B
**2022**, 105, 195130. [Google Scholar] [CrossRef] - Rycerz, A.; Spałek, J. Exact Diagonalization of Many-Fermion Hamiltonian with Wave-Function Renormalization. Phys. Rev. B
**2001**, 63, 073101. [Google Scholar] [CrossRef] - Spałek, J.; Rycerz, A. Electron localization in one-dimensional nanoscopic system: A combined exact diagonalization-an ab initio approach. Phys. Rev. B
**2001**, 64, 161105. [Google Scholar] [CrossRef] [Green Version] - Singha, A.; Gibertini, M.; Karmakar, B.; Yuan, S.; Polini, M.; Vignale, G.; Katsnelson, M.I.; Pinczuk, A.; Pfeiffer, L.N.; West, K.W.; et al. Two-dimensional Mott-Hubbard electrons in an artificial honeycomb lattice. Science
**2011**, 332, 1176. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Polini, M.; Guinea, F.; Lewenstein, M.; Manoharan, H.C.; Pellegrini, V. Artificial honeycomb lattices for electrons, atoms and photons. Nat. Nanotechnol.
**2013**, 8, 625. [Google Scholar] [CrossRef] [Green Version] - Gardenier, T.S.; van den Broeke, J.J.; Moes, J.R.; Swart, I.; Delerue, C.; Slot, M.R.; Smith, C.M.; Vanmaekelbergh, D. p Orbital Flat Band and Dirac Cone in the Electronic Honeycomb Lattice. ACS Nano
**2020**, 14, 13638. [Google Scholar] [CrossRef] - Trainer, D.J.; Srinivasan, S.; Fisher, B.L.; Zhang, Y.; Pfeiffer, C.R.; Hla, S.-W.; Darancet, P.; Guisinger, N.P. Manipulating topology in tailored artificial graphene nanoribbons. arXiv
**2021**, arXiv:2104.11334. [Google Scholar] - Cao, Y.; Fatemi, V.; Demir, A.; Fang, S.; Tomarken, S.L.; Luo, J.Y.; Sanchez-Yamagishi, J.D.; Watanabe, K.; Taniguchi, T.; Kaxiras, E.; et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature
**2018**, 556, 80. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fidrysiak, M.; Zegrodnik, M.; Spałek, J. Unconventional topological superconductivity and phase diagram for an effective two-orbital model as applied to twisted bilayer graphene. Phys. Rev. B
**2018**, 98, 085436. [Google Scholar] [CrossRef] [Green Version] - Lee, S.-H.; Chung, H.-J.; Heo, J.; Yang, H.; Shin, J.; Chung, U.-I.; Seo, S. Band Gap Opening by Two-Dimensional Manifestation of Peierls Instability in Graphene. ACS Nano
**2011**, 5, 2964. [Google Scholar] [CrossRef] - Lee, S.-H.; Kim, S.; Kim, K. Semimetal-antiferromagnetic insulator transition in graphene induced by biaxial strain. Phys. Rev. B
**2012**, 86, 155436. [Google Scholar] [CrossRef] [Green Version] - Sorella, S.; Seki, K.; Brovko, O.O.; Shirakawa, T.; Miyakoshi, S.; Yunoki, S.; Tosatti, E. Correlation-Driven Dimerization and Topological Gap Opening in Isotropically Strained Graphene. Phys. Rev. Lett.
**2018**, 121, 066402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Eom, D.; Koo, J.-Y. Direct measurement of strain-driven Kekulé distortion in graphene and its electronic properties. Nanoscale
**2020**, 12, 19604. [Google Scholar] [CrossRef] [PubMed] - Bao, C.; Zhang, H.; Zhang, T.; Wu, X.; Luo, L.; Zhou, S.; Li, Q.; Hou, Y.; Yao, W.; Liu, L.; et al. Exp. Evid. Chiral Symmetry Break. Kekulé-Ordered Graphene. Phys. Rev. Lett.
**2021**, 126, 206804. [Google Scholar] [CrossRef] - Costa, N.C.; Seki, K.; Sorella, S. Magnetism and Charge Order in the Honeycomb Lattice. Phys. Rev. Lett.
**2021**, 126, 107205. [Google Scholar] [CrossRef] - Dresselhaus, G.; Dresselhaus, M.S.; Saito, R. Physical Properties of Carbon Nanotubes; World Scientific: Singapore, 1998; Chapter 11. [Google Scholar] [CrossRef]
- Tsai, J.-L.; Tu, J.-F. Characterizing mechanical properties of graphite using molecular dynamics simulation. Mater. Des.
**2010**, 31, 194. [Google Scholar] [CrossRef] - Hur, K.L. Weakly coupled Hubbard chains at half-filling and confinement. Phys. Rev. B
**2001**, 63, 165110. [Google Scholar] [CrossRef] [Green Version] - Spałek, J.; Görlich, E.M.; Rycerz, A.; Zahorbeński, R. The combined exact diagonalization-ab initio approach and its application to correlated electronic states and Mott-Hubbard localization in nanoscopic systems. J. Phys. Condens. Matter
**2007**, 19, 255212. [Google Scholar] [CrossRef] - Lenz, B.; Manmana, S.R.; Pruschke, T.; Assaad, F.F.; Raczkowski, M. Mott Quantum Criticality in the Anisotropic 2D Hubbard Model. Phys. Rev. Lett.
**2016**, 116, 086403. [Google Scholar] [CrossRef] - Due to mirror symmetries, it sufficient to sum over first quarter of the hexagonal Brilloun zone, namely, for 0 ⩽ k
_{y}< 2π/$\sqrt{3}$ and 0 ⩽ k_{x}< 4π/3−|k_{y}|/$\sqrt{3}$. - Castro Neto, A.H.; Guinea, F.; Peres, N.M.R.; Novoselov, K.S.; Geim, A.K. The electronic properties of graphene. Rev. Mod. Phys.
**2009**, 81, 109. [Google Scholar] [CrossRef] [Green Version] - Typically, for each combination of t
_{y}/t_{x}, U/t_{x}and Δ/t_{x}, we took 15÷20 values of the parameter g = e^{−η}separated by the steps of 0.01, in the vicinity of a predicted energy minimum. For each g, the averages over distributions of electrons in real space were calculated by performing 10^{5}iterations per lattice, according to the Glauber’s algorithm [see, e.g.: M. Lewerenz, Monte Carlo Methods: Overview and Basics. In: Grotendorst, J.; Marx, D.; Muramatsu, A. (eds) Quantum simulations of complex many-body systems: From theory to algorithms, lecture notes; winter school, 25 February–1 March 2002, Rolduc Conference Centre, Kerkrade, The Netherlands. John von Neumann Institute for Computing Jülich, 2002. http://hdl.handle.net/2128/2921]. Initial 10^{4}iterations per site was neglected for each simulation, to avoid the effects of initial configuration. The variational energy ${E}_{G}^{\mathrm{(GWF)}}$ together with the corresponding optimal value of the parameter g were then determined via the least-squares fitting of a quadratic function. - Takano, F.; Uchinami, M. Application of the Gutzwiller Method to Antiferromagnetism. Prog. Theor. Phys.
**1975**, 53, 1267. [Google Scholar] [CrossRef] [Green Version] - Vollhardt, D. Normal
^{3}He: An almost localized Fermi liquid. Rev. Mod. Phys.**1984**, 56, 99. [Google Scholar] [CrossRef] - Jędrak, J.; Kaczmarczyk, J.; Spałek, J. Statistically-consistent Gutzwiller approach and its equivalence with the mean-field slave-boson method for correlated systems. arXiv
**2010**, arXiv:1008.0021. [Google Scholar] - Lanatà, N.; Strand, H.U.R.; Dai, X.; Hellsing, B. Efficient implementation of the Gutzwiller variational method. Phys. Rev. B
**2012**, 85, 035133. [Google Scholar] [CrossRef] [Green Version] - Wysokiński, M.M.; Spałek, J. Properties of an almost localized Fermi liquid in an applied magnetic field revisited: A statistically consistent Gutzwiller approach. J. Phys. Condens. Matter
**2014**, 26, 055601. [Google Scholar] [CrossRef] [PubMed] - Chern, G.-W.; Barros, K.; Batista, C.D.; Kress, J.D.; Kotliar, G. Mott Transition in a Metallic Liquid: Gutzwiller Molecular Dynamics Simulations. Phys. Rev. Lett.
**2017**, 118, 226401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fidrysiak, M.; Zegrodnik, M.; Spałek, J. Realistic estimates of superconducting properties for the cuprates: Reciprocal-space diagrammatic expansion combined with variational approach. J. Phys. Condens. Matter
**2018**, 30, 475602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gutzwiller, M.C. Effect of Correlation on the Ferromagnetism of Transition Metals. Phys. Rev.
**1964**, 134, A923. [Google Scholar] [CrossRef] - Gutzwiller, M.C. Correlation of Electrons in a Narrow s Band. Phys. Rev.
**1965**, 137, A1726. [Google Scholar] [CrossRef] - Kennedy, T.; Lieb, E.H.; Shastry, B.S. The XY Model Has Long-Range Order for All Spins and All Dimensions Greater than One. Phys. Rev. Lett.
**1988**, 61, 2582. [Google Scholar] [CrossRef] - Pereira, V.M.; Castro Neto, A.H.; Peres, N.M.R. Tight-binding approach to uniaxial strain in graphene. Phys. Rev. B
**2009**, 80, 045401. [Google Scholar] [CrossRef] [Green Version] - Tran, M.-T.; Kuroki, K. Finite temperature semimetal insulator transition on the honeycomb lattice. Phys. Rev. B
**2009**, 79, 125125. [Google Scholar] [CrossRef] [Green Version] - Capello, M.; Becca, F.; Fabrizio, M.; Sorella, S.; Tosatti, E. Variational Description of Mott Insulators. Phys. Rev. Lett.
**2005**, 94, 026406. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Biborski, A.; Kądzielawa, A.P.; Spałek, J. Atomization of correlated molecular-hydrogen chain: A fully microscopic variational Monte Carlo solution. Phys. Rev. B
**2018**, 98, 085112. [Google Scholar] [CrossRef] [Green Version] - Gröning, O.; Wang, S.; Yao, X.; Pignedoli, C.A.; Barin, G.B.; Daniels, C.; Cupo, A.; Meunier, V.; Feng, X.; Narita, A.; et al. Eng. Robust Topol. Quantum Phases Graphene Nanoribbons. Nature
**2018**, 560, 209. [Google Scholar] [CrossRef] [Green Version] - Rycerz, A. Strain-induced transitions to quantum chaos and effective time-reversal symmetry breaking in triangular graphene nanoflakes. Phys. Rev. B
**2013**, 87, 195431. [Google Scholar] [CrossRef] [Green Version] - Rostami, H.; Asgari, R. Electronic ground-state properties of strained graphene. Phys. Rev. B
**2012**, 86, 155435. [Google Scholar] [CrossRef] [Green Version] - Oliva-Leyva, M.; Naumis, G.G. Generalizing the Fermi velocity of strained graphene from uniform to nonuniform strain. Phys. Lett. A
**2015**, 379, 2645. [Google Scholar] [CrossRef] [Green Version] - Singh, J.; Jamdagni, P.; Jakhara, M.; Kumar, A. Stability, electronic and mechanical properties of chalcogen (Se and Te) monolayers. Phys. Chem. Chem. Phys.
**2020**, 22, 5749. [Google Scholar] [CrossRef] - Zhang, G.; Lu, K.; Wang, Y.; Wang, H.; Chen, Q. Mechanical and electronic properties of α−M
_{2}X_{3}(M = Ga, In; X = S, Se) monolayers. Phys. Rev. B**2022**, 105, 235303. [Google Scholar] [CrossRef]

**Figure 1.**

**Top**: Honeycomb lattice subjected to uniaxial strain in selected direction (see the coordinate system). Zoom-in visualizes the distance (bond length) ${d}_{ij}$ between atoms i and j and in-plane angle with the vertex at site j ($\measuredangle \left(j\right)$).

**Bottom**: Hexagonal first Brillouin zone (FBZ) of the reciprocal lattice, with (dimensionless) basis vectors ${\mathbf{b}}_{1}=\left(2\pi /\sqrt{3}\right)\phantom{\rule{0.166667em}{0ex}}(\sqrt{3},-1)$ and ${\mathbf{b}}_{\mathbf{2}}=\left(2\pi /\sqrt{3}\right)\phantom{\rule{0.166667em}{0ex}}(0,2)$, and the symmetry points, $\mathbf{K}=(4\pi /3,0)$ and ${\mathbf{K}}^{\prime}=(2\pi /3,2\pi /\sqrt{3})$ coinciding with Dirac points in the absence of strain. The magnified area shows discretized FBZ for a finite system of $N=2{N}_{x}{N}_{y}$ atoms with periodic boundary conditions [see Equation (6)]. (The values of ${N}_{x}=4$ and ${N}_{y}=5$ are used for illustration only).

**Figure 2.**Density of states for the Hamiltonian (1) with $U=0$ displayed as a function of energy. Top: strain applied in the armchair direction (${t}_{y}\u2a7d{t}_{x}$). Bottom: strain applied in the zigzag direction (${t}_{y}\u2a7e{t}_{x}$). The ratio ${t}_{<}/{t}_{>}$ [with ${t}_{<}=\mathrm{min}\phantom{\rule{0.166667em}{0ex}}({t}_{x},{t}_{y})$ and ${t}_{>}=\mathrm{max}\phantom{\rule{0.166667em}{0ex}}({t}_{x},{t}_{y})$] is varied between the lines with the steps of $0.2$. A vertical offset is applied to each dataset except from the isotropic case (${t}_{x}={t}_{y}$). Inset shows the band gap, appearing for ${t}_{x}<0.5\phantom{\rule{0.166667em}{0ex}}{t}_{y}$ due to the Peierls transition.

**Figure 3.**(

**a**–

**d**) Main: Energy difference between the antiferromagnetic Gutzwiller variational energy ${E}_{G}^{\left(\mathrm{GWF}\right)}\left(m\right)$ [see Equation (13)] and the paramagnetic solution ${E}_{G}^{\left(\mathrm{GWF}\right)}\left(0\right)$ obtained from VMC simulations as a function the on-site Hubbard repulsion (U). The parameter $\eta $ is optimized for a fixed $m=\Delta /U$ (or $m=0$); $\Delta $ is varied between the lines from $\Delta /{t}_{x}=0.25$ to $\Delta /{t}_{x}=1$, with the steps of $0.25$. The hopping anisotropy ${t}_{y}/{t}_{x}$ is varied between the panels. Inset shows the value of $U={U}_{0}(\Delta )$ at which $\Delta {E}_{G}^{\left(\mathrm{GWF}\right)}$ changes sign for a given $\Delta $. The extrapolation to $\Delta \to 0$ yields the critical values of ${U}_{\mathrm{c}}^{\left(\mathrm{GWF}\right)}$ given in Table 1. (Statistical errorbars are too small to be shown on the plots).

**Figure 4.**(

**a**–

**d**) The lower and the upper bounds to the critical Hubbard repulsion ${U}_{c}$ for anisotropic honeycomb lattice estimated by comparing different versions of the Gutzwiller Approximation described in the text. The lower bound (${U}_{c}^{\left(\mathrm{GA}\right)}$) coincides with the splitting of the Gutzwiller energy for paramagnetic state, ${E}_{G}^{\left(\mathrm{GA}\right)}(m\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0)$, given by Equation (19) [red dashed line] and the variational energy ${E}_{G}^{\left(\mathrm{GA}\right)}$, see Equation (17), with the parameters $(m,d)$ optimized numerically [blue solid line], both displayed as functions of U. The value of ${U}_{c}^{\left(\mathrm{GA}\right)}$ is obtained via the extrapolation with $m\to 0$, similarly as for the VMC results in Figure 3. The intersection of ${E}_{G}^{\left(\mathrm{GA}\right)}(m\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0)$ with ${E}_{G}^{\mathrm{NGA}}$, see Equation (20) [green dashed-dotted line] yields the upper bound (${U}_{c}^{\left(\mathrm{NGA}\right)}$). The value of the ${t}_{y}/{t}_{x}$ ratio is varied between the panels. [For the numerical values of ${U}_{c}^{\left(\mathrm{GA}\right)}$ and ${U}_{c}^{\left(\mathrm{NGA}\right)}$, see Table 1].

**Figure 5.**

**Top**: Phase diagram for the Hubbard model on anisotropic honeycomb lattice, see Equations (1) and (2), with ${t}_{y}\u2a7d{t}_{x}$ corresponding a gapless single-particle spectrum, see Figure 2. [Here, ${t}_{>}={t}_{x}$ and ${t}_{<}={t}_{y}$.] Lines depict the critical Hubbard repulsion estimated within the Hartree–Fock method [short dashed], Gutzwiller Approximation [thick solid], Statistically Consistent GA [long dashed-double dotted], Coherent Potential Approximation [dotted], and the Néel-state GA [long dashed-dotted]. Datapoints with errorbars are obtained from VMC simulations for the Gutzwiller Wave Function, see Equation (12); thin dash-dotted line represents a power-law fit given by Equation (23) with thin solid lines bounding the statistical uncertainty (yellow area). Quantum Monte Carlo value for the isotropic case, ${U}_{c}/{t}_{0}=3.86$ [18], is also marked (full circle).

**Bottom**: A zoom in, with trajectories following from the SSH model for strained graphene (see Appendix B) for $\beta =2$ (red line/open symbols) and $\beta =3$ (blue line/closed symbols). Different datapoints for each value of $\beta $ correspond to the applied strain ${\epsilon}_{y}$ varied from ${\epsilon}_{y}=0.05$ to $0.25$ with the steps of $0.05$. (GA and VMC results are omitted for clarity.) Remaining Labels/colored areas: the semimetallic phase (SM) [blue] with the correlated-semimetal range (CSM) [light blue] and the Mott insulator (MI) [magenta].

**Figure 6.**The evolution of critical Hubbard interaction with armchair strain strain (${t}_{y}\u2a7d{t}_{x}$), approximated by Equation (26) [solid lines] for two values of ${U}_{c}^{\left(0\right)}$ (specified on the plot) adjusted to match the zero-strain results obtained from HF and GWF methods. The remaining lines are the same as in Figure 5.

**Figure 7.**(

**a**–

**c**) Average kinetic energy per site and (

**d**–

**f**) average double occupancy displayed as functions of the Hubbard repulsion U for ${t}_{y}/{t}_{x}=1$ (top), ${t}_{y}/{t}_{x}=0.75$ (middle), and ${t}_{y}/{t}_{x}=0.5$ (bottom). Thick dashed line marks the Hartree–Fock results, thick solid line represents the Gutzwiller Approximation. Datapoints depict the VMC results for GWF with $m=0$ (red open symbols) and optimized m (blue closed symbols); thin lines are guide for the eye only. Shaded area marks the correlated semimetallic phase, bounded by ${U}_{c}^{\left(\mathrm{HF}\right)}$ and ${U}_{c}^{\left(\mathrm{GWF}\right)}$. (For the numerical values, see Table 1).

**Table 1.**Critical values of the Hubbard repulsion ${U}_{c}^{\left(\mathrm{GWF}\right)}$ obtained from VMC simulations (with standard deviations for the last digit specified in parentheses) compared with the upper (${U}_{c}^{\left(\mathrm{GA}\right)}$) and the upper (${U}_{c}^{\left(\mathrm{NGA}\right)}$) bound following from the Gutzwiller Approximation (GA) and the Néel-state Gutzwiller Approximation (NGA). The results obtained from the Statistically-consistent Gutzwiller Approximation (SGA) are also given. The system size is defined by ${N}_{x}={N}_{y}=10$ for VMC simulations; the remaining results correspond to the limit of ${N}_{x}={N}_{y}\to \infty $.

$\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathit{t}}_{\mathit{y}}/{\mathit{t}}_{\mathit{x}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ | $\phantom{\rule{4pt}{0ex}}{\mathit{U}}_{\mathit{c}}^{\left(\mathbf{GWF}\right)}\phantom{\rule{-0.166667em}{0ex}}/{\mathit{t}}_{\mathit{x}}\phantom{\rule{4pt}{0ex}}$ | $\phantom{\rule{4pt}{0ex}}{\mathit{U}}_{\mathit{c}}^{\left(\mathbf{GA}\right)}\phantom{\rule{-0.166667em}{0ex}}/{\mathit{t}}_{\mathit{x}}\phantom{\rule{4pt}{0ex}}$ | $\phantom{\rule{4pt}{0ex}}{\mathit{U}}_{\mathit{c}}^{\left(\mathbf{SGA}\right)}\phantom{\rule{-0.166667em}{0ex}}/{\mathit{t}}_{\mathit{x}}\phantom{\rule{4pt}{0ex}}$ | $\phantom{\rule{4pt}{0ex}}{\mathit{U}}_{\mathit{c}}^{\left(\mathbf{NGA}\right)}\phantom{\rule{-0.166667em}{0ex}}/{\mathit{t}}_{\mathit{x}}\phantom{\rule{4pt}{0ex}}$ |
---|---|---|---|---|

1.00 | 3.48(1) | 2.804 | 3.12 | 5.281 |

0.75 | 2.91(1) | 2.550 | 2.83 | 4.871 |

0.50 | 2.69(3) | 2.241 | 2.47 | 4.508 |

0.25 | 2.24(1) | 1.830 | 1.98 | 4.199 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rut, G.; Fidrysiak, M.; Goc-Jagło, D.; Rycerz, A.
Mott Transition in the Hubbard Model on Anisotropic Honeycomb Lattice with Implications for Strained Graphene: Gutzwiller Variational Study. *Int. J. Mol. Sci.* **2023**, *24*, 1509.
https://doi.org/10.3390/ijms24021509

**AMA Style**

Rut G, Fidrysiak M, Goc-Jagło D, Rycerz A.
Mott Transition in the Hubbard Model on Anisotropic Honeycomb Lattice with Implications for Strained Graphene: Gutzwiller Variational Study. *International Journal of Molecular Sciences*. 2023; 24(2):1509.
https://doi.org/10.3390/ijms24021509

**Chicago/Turabian Style**

Rut, Grzegorz, Maciej Fidrysiak, Danuta Goc-Jagło, and Adam Rycerz.
2023. "Mott Transition in the Hubbard Model on Anisotropic Honeycomb Lattice with Implications for Strained Graphene: Gutzwiller Variational Study" *International Journal of Molecular Sciences* 24, no. 2: 1509.
https://doi.org/10.3390/ijms24021509