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

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## 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

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**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 |

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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