Inducing and controlling superconductivity in Hubbard honeycomb model using an electromagnetic drive

Graphene has been one of the most studied materials with hexagonal geometry due to its number of intriguing properties Novoselov et al. (2004); Geim and Novoselov (2007); Castro Neto et al. (2009). Recently, superconductivity was observed in a twisted bilayer at the magic angle Cao et al. (2018), after the theoretical prediction of the existence of flat bands for Moiré lattice of the twisted bilayer graphene at the magic angle Bistritzer and MacDonald (2011). Superconductivity has also been observed in intercalated graphite such as ${\mathrm{CaC}}_{6}$ Emery et al. (2005), and C₆Yb Weller et al. (2005). But, in pristine graphene, superconductivity is still missing in spite of a number of theoretical studies predicting the existence of superconductivity Black-Schaffer and Doniach (2007); Honerkamp (2008); Profeta et al. (2012); Nandkishore et al. (2012). Ref. Nandkishore et al. (2012) predicted chiral superconductivity with nontrivial topology at van Hove singularity, which was an experimental challenge. More recently, doping graphene at and beyond van Hove singularity the have been achieved Rosenzweig et al. (2020) and can possibly open new venues for exotic states. Many of these studies start with the assumption that honeycomb lattice has is a spin-$1/2$ antiferromagnet at half-filling, which has, in fact been observed in another honeycomb material, In₃Cu₂VO₉ Möller et al. (2008); Yan et al. (2012). More recently, valence bond fluctuations were reported in another spin-1/2 honeycomb compound, YbBr₃ Wessler et al. (2020). On the other hand, tunable honeycomb lattice in the cold atom system has been realized Tarruell et al. (2012).

In recent years, there has also been an enormous interest in understanding the periodically driven systems. A recent study on graphene McIver et al. (2020) reported the light-induced anomalous quantum Hall effect by tuning its band structure using light  Oka and Aoki (2009). Controlling the magnetic interaction in materials using an electromagnetic drive has been intensively studied Beaurepaire et al. (1996); Mentink et al. (2015, 2012); Kirilyuk et al. (2010); Li et al. (2013); Oka and Kitamura (2019). The earlier work on photo-manipulation has largely focused on ferromagnets. But in recent years, study of antiferromagnets driven by light have also gained significant interest Nemec et al. (2018).

A recent study Dehghani and Mitra (2017) on hexagonal lattice proposed the realization of light-induced time-reversal symmetry broken (TRSB) topological superconductor. In this study, the effect of electromagnetic drive was studied using a Pierls substitution and did not delve into how the frequency restricts the realization of the relevant Floquet $t$-$J$ model. Also, in one of the most studied superconductors, $i.e$ cuprate, there has been significant interest in enhancing superconductivity in the presence of a drive Fausti et al. (2011); Mankowsky et al. (2014) resonant with phonon mode, and the mechanism it is usually understood as phonon-assisted. More recently Liu et al. (2020), it has been shown that the enhancement in the superconductivity can be observed even when one is away from the resonance with the phonon. The mechanism, in this case, has been attributed to the effect of light on the superexchange coupling.

The observation of band tuning in graphene McIver et al. (2020) and the possibility of light-induced superconductivity mediated by charge transfer Liu et al. (2020) opens new avenues for exploring light-induced superconductivity. Additionally, it has been widely reported in the literature that strong correlations can induce superconductivity in honeycomb lattice at low doping Black-Schaffer and Honerkamp (2014); Wu et al. (2013) and van Hove singularity Nandkishore et al. (2012). In our work, we explore the conditions for realizing superconductivity in honeycomb lattice mediated by strong correlations that can be controlled using light. We explore the effect of both the circularly and linearly polarized light on the electronic states. We show that in the frequency limit ($t\ll\omega\ll U$. Here $t$ is the hopping strength and $U$ is the onsite Coulomb interaction.), the superconductivity can be induced and enhanced. The honeycomb lattice is shown to host both the time reversal symmetry breaking (TRSB) ($d+id$) and the nematic ($s+d$) superconductor in the singlet pairing channel. The nematic superconducting order parameter is three-fold degenerate. We further show that the three-fold degeneracy can be lifted using linearly polarized light.

The paper is organized as follows: Sec II presents the Floquet $t$-$J$ model derived from Hubbard honeycomb model in the presence of circularly and linearly polarized light. Sec. III discusses the density of states for the lattice in the presence of drive. Sec. IV discusses the mean-field treatment of Floquet $t$-$J$ model, which reveals several superconducting phases in the presence of EM drive. Sec. V presents a discussion on the physical challenges in realizing the these Floquet systems and summarizes the various findings of our work.

Strongly correlated materials can be modeled using Hubbard model. Time-dependence of the Hamiltonian in the presenc of a drive can be is taken into account by time-periodic Peierls phase Dehghani and Mitra (2017); Wang et al. (2018); Eckardt and Anisimovas (2015). We start with a time-dependent Hamiltonian for a Hubbard honeycomb lattice given by

Here, $t_{ij}$ is the hopping between nearest neighbor sites-$i,j$, and $U$ is the onsite electron repulsion. $\delta F_{ij}(t)=\textbf{F}(t)\cdot(\textbf{r}_{i}-\textbf{r}_{j})$, where $\textbf{F}(t)$ is the vector potential of the light. The above time-dependent Hubbard Hamiltonian can be mapped onto a Floquet $t$-$J$ model dependent using Schrieffer-Wolff Transformation (SWT) Bukov et al. (2016); MacDonald et al. (1988). The details for the SWT are discussed in the supplemental material Sup .

We investigate the Floquet $t$-$J$ model for two types of drive, $\textbf{ F}(t)$: a) circularly polarized light (CPL) and b) linearly polarized light (LPL) in the supplemental material Sup . The time-dependent Hamiltonian in the high frequency limit ($t/\omega\ll 1$), and $t\ll U$ can be mapped onto a set of time-independent Floquet $t$-$J$ Hamiltonians depending on the drive frequency ($\omega$) Eckardt and Anisimovas (2015); Bukov et al. (2016). Here, we present the Floquet Hamiltonian for both circularly and linearly polarized light in the discussion below.

Circularly Polarized Light:— We consider a vector potential given by $\textbf{F}(t)=\zeta[\sin(\omega t)\hat{e}_{x}-\cos(\omega t)\hat{e}_{y}]$, where $\zeta=A/\omega$ for the circularly polarized light (CPL) and $\hat{e}_{x}$ ($\hat{e}_{y}$) is a unit vector in the $x$ ($y$) direction. In this case, the $D_{\text{6h}}$ point group symmetry of the Hamiltonian is preserved. This is because the CPL affects both the hopping and interaction isotropically along the three bonds in the hexagonal lattice. We have two sets of Floquet Hamiltonian depending on the conditions on the frequency.

In the limit $t\ll U\ll\omega$, the Floquet Hamiltonian is given by

Here, $\tilde{c}_{j\sigma}=(1-n_{j\bar{\sigma}})c_{j\sigma}$, $\tilde{n}_{j\sigma}=(1-n_{j\bar{\sigma}})n_{j\sigma}$ and $J=\frac{4t^{2}}{U}$ is the superexchange interaction. $\mathcal{J}_{0}(\zeta)$ is a Bessel function of first kind with $\zeta=A/\omega$. In this limit of $\omega$, both the hopping and superexchange are rescaled in the Floquet Hamiltonian as shown in the panel (b) of Fig. 1.

On the other hand in the limit $t\ll\omega\ll U$, one can write the Floquet Hamiltonian as

In this limit, only the hopping term is rescaled, whereas the superexchange term remains unchanged. This limit can allows for flattening the non-interacting band and whereas the superexchange term interaction is constant, and hence an ideal condition for inducing and enhancing superconductivity.

Additionally, we have chiral next-nearest neighbor (NNN) hopping terms of the order of $1/\omega$ that can be neglected owing to high-frequency approximation. Further, there are additional three site hopping terms that contribute at the same order as superexchange in the high-frequency approximation of the Floquet theory MacDonald et al. (1988); Bukov et al. (2016). We further comment on the conditions under which these extra terms can be neglected in the supplemental Sup .

Linearly Polarized Light:— We consider a linearly polarized light (LPL) polarized along the $y$-direction for which the vector potential is given by $\textbf{F}(t)=\zeta\sin(\omega t)\hat{e}_{y}$, where $\zeta=A/\omega$. In this case, drive breaks the $C_{\text{3}}$-rotation symmetry along the three bonds in the Hamiltonian, and can lead to anisotropic hopping and interactions. As in the CPL case, here too, the Hamiltonian depends on the $U$ and $\omega$ strength.

In the limit $t\ll U\ll\omega$ limit, the Floquet Hamiltonian is given by

In this case, both the hopping and superexchange are rescaled as is the case in CPL drive. In addition, the system develops anisotropy along the three bonds due to the explicit $C_{3}$ rotation symmetry breaking by the LPL.

On the other hand in the limit $t\ll\omega\ll U$, the Hamiltonian is given by

In this case, the hopping is rescaled and develop anisotropy, whereas the superexchange is unaffected. It leads to flattening of the non-interacting band, whereas the superexchange term interaction is constant, and hence is useful for inducing and enhancing superconductivity

As in the CPL case, we do have NNN hopping terms, but the chiral hopping term is absent in the LPL drive as the Hamiltonian does not break time-reversal symmetry.

The effect of EM drive on non-interacting honeycomb lattice has been widely studied in literature, where the EM drive was found to have significant effects on electronic properties Oka and Aoki (2009). Here, we analyze the impact of drive on the honeycomb lattice without the superexchange term to prepare ourselves for understanding the more complicated Hamiltonian with strong correlations. We report the density of states (DOS) for non-interacting bands in the presence of both circularly and linearly polarized light.

Circularly polarized light:— In the case of CPL, the hopping is renormalized isotropically and, therefore, preserves the $C_{\text{3}}$-rotation symmetry of the bonds. Panel (a) in Fig. 2 shows the density of states for CPL drive. We can see that on driving the system ($\zeta$), only the bandwidth in the density of states shown in panel (a) of Fig. 2 gets smaller.

Linearly polarized light:— Linearly polarized light can induce anisotropy in the hopping along with different bonds. It breaks $C_{\text{3}}$-rotation symmetry of the bonds and can generate effects on the honeycomb sites similar to the strain Naumis et al. (2017). Panel (b) shows the DOS for the LPL.

Additionally, CPL can breaks time-reversal symmetry in the Floquet $t$-$J$ model if the next nearest neighbor chiral hoppings are included. This term can lead to an anomalous quantum Hall effect (AQHE). The observation of light-induced AQHE in monolayer graphene was reported very recently McIver et al. (2020). This TRSB breaking term appears as a higher-order correction, and is only important when the leading order term proportional to the zeroth order Bessel function vanishes, $i.e.$ at $A/\omega=2.4$ Mikami et al. (2016). For the sake of simplicity, we study the effects of drive in the limit of $\zeta<2$, which allows us to ignore the TRSB term.

In Sec. II, we have presented the Floquet Hamiltonian that can be engineered from a honeycomb Hubbard model. The derived $t$-$J$ Floquet model has an explicit many-body effect, which cannot be solved exactly. To simplify the many-body effects, one can use Gutzwiller projection to replace the strict double occupancy prohibition with a rescaling factor Zhang et al. (1988); Anderson et al. (2004); Paramekanti et al. (2004); Black-Schaffer and Honerkamp (2014). The Gutzwiller projected Hamiltonian could then be mapped to renormalized mean-field theory, using the mean-field for singlet pairing channel. In this renormalized mean-field theory, $t\rightarrow 2t\delta/(1+\delta)$ and $J\rightarrow 2J/(1+\delta)^{2}$ and superconducting order parameter, $\Delta\rightarrow 2\Delta\delta/(1+\delta)$, where $\delta~{}(=1-n)$ is the doping away from half-filling. For the sake of simplicity, we will treat these rescaling factors as a constant. The Gutzwiller projected mean-field can be written as,

Here, $t_{ij}$ is the hopping between nearest neighbor sublattices, $A~{}(a_{i\sigma})$ and $B~{}(b_{i\sigma})$. Here we restrict to the nearest neighbor singlet pairing superconducting order parameter (SCOP) $\Delta_{ij}=-J_{ij}(a_{i\downarrow}b_{j\uparrow}-a_{i\uparrow}b_{j\downarrow})/2$.

Equation  (6) can be written in the momentum basis as

Here $t_{\textbf{k}}=\sum_{\alpha=0,1,2}e^{i\textbf{k}\cdot\textbf{l}_{\alpha}}t_{\alpha}$ and $\Delta_{\textbf{k}}=\sum_{\alpha=0,1,2}e^{i\textbf{k}\cdot\textbf{l}_{\alpha}}\Delta_{\alpha}$, where $\textbf{R}_{j}=\textbf{R}_{i}+\textbf{l}_{\alpha}$ with $l_{0}=(0,-1),~{}l_{1}=(\frac{\sqrt{3}}{2},\frac{1}{2}),~{}l_{2}=(-\frac{\sqrt{3}}{2},\frac{1}{2})$. Also, one can write the gap as, $\Delta_{\alpha}=-\frac{J_{\alpha}}{2}\sum_{k}\langle e^{i\textbf{k}\cdot\textbf{l}_{\alpha}}a_{-k\downarrow}b_{k\uparrow}-e^{-i\textbf{k}\cdot\textbf{l}_{\alpha}}a_{k\uparrow}b_{-k\downarrow}\rangle$. Equation (7) can be diagonalized to evaluate $H_{\text{MF}}|\psi_{m}\rangle=E_{m}|\psi_{m}\rangle$, the eigenstates ($|\psi_{m}\rangle$) of which are the Bogoliubov quasiparticles  Su and Lin (2018). The gap equation in this new basis is then given by

Here, $|\psi_{k}^{m}\rangle=\big{(}u_{k\uparrow}^{a,m},~{}v_{k\uparrow}^{b,m},~{}(u_{-k\downarrow}^{a,m})^{*},~{}(v_{-k\downarrow}^{b,m})^{*}\big{)}^{T}$ and $E_{k}^{m}$ are the $m^{\text{th}}$ eigenvector and eigenvalue of the Eq. (7). $f(E_{k}^{m})=\langle(\gamma_{k}^{m})^{\dagger}\gamma_{k}^{m}\rangle=\frac{1}{e^{\beta E_{k}^{m}}+1}$ is the Fermi distribution function for the Bogoliubov quasiparticles. We solve Eqs. (8) and (9) self-consistently to evaluate the SC order parameter.

The basis functions; $\Psi_{d_{x^{2}-y^{2}}}=\tfrac{1}{\sqrt{6}}(2,-1,-1)$, $\Psi_{d_{xy}}=\tfrac{1}{\sqrt{2}}(0,1,-1)$ and $\Psi_{s}=\tfrac{1}{\sqrt{3}}(1,1,1)$ form a complete basis set for symmetric superconducting order parameter ($\Delta_{\alpha}$) for point group $D_{\text{6h}}$ and $D_{\text{2h}}$. Moreover, $d_{x^{2}-y^{2}}$ and $d_{xy}$ are degenerate for $D_{\text{6h}}$ point group, which allows for the existence of interesting chiral superconductivity $d_{x^{2}-y^{2}}+id_{xy}$. We evaluate the SCOP in these basis functions. All throughout the discussion, we set $t=1$ and all other parameters are defined in terms of $t$.

In our work, we explored the conditions for enhancing superconductivity in strongly correlated honeycomb lattice in the presence of both circularly and linearly polarized light. An earlier work Dehghani and Mitra (2017) on honeycomb lattice also explored the same system by treating EM drive through a Pierls-substitution. On the other hand, we have treated the effect of strong correlation systematically using SWT and find that the limit $t\ll\omega\ll U$ is useful in enhancing superconductivity. We want to point out that two conditions are important for realizing such enhancement; a) $t\ll\omega\ll U$, such that the higher-order corrections in the high-frequency approximations can be neglected, and b) thermal heating is minimal to realize Floquet Hamiltonian. To have a minimal effect on the higher-order correction, we study the system for $\zeta<2$. In driven systems, a number of studies have reported the effect of heating and shown that, indeed, it can be controlled. For example, it has been shown that the Floquet heating in the many-body systems in the high frequency $t,U<\omega$ is exponentially slow in frequency  Abanin et al. (2015). Further, it has been shown that Floquet prethermalized state can be realized in a resonantly driven Hubbard model Herrmann et al. (2017); Coulthard et al. (2017). Additionally, the recent experimental observation of off-resonance superconductivity enhancement in cuprate Liu et al. (2020), possibly mediated by charge fluctuations, provides further impetus to our work.

In conclusion, we have shown that light can be a useful tool for controlling the bandwidth and strong correlations in a hexagonal lattice. We show that in the frequency limit, $t<\omega$ and $\omega<U$, the EM field drive can induce and enhance superconductivity mediated through strong correlations. We show that light-induced superconductivity is exotic; a) time-reversal symmetry breaking ($d+id$) with nontrivial topology and b) nematic ($s+d$) in these honeycomb antiferromagnets. We also show how the TRSB state can be driven into a nematic superconductor in the presence of both; circularly and linearly polarized light. Our study also explores the effect of temperature on the SCOP and presents a phase diagram for the temperature dependence of the SCOP in the presence of a CPL drive. On the discussion of the temperature of dependence, we also present all the three-fold degenerate nematic state and confirm their degeneracy using a calculation of the free energy. We further show that this three-fold degeneracy of the nematic state can be reduced to a two-fold in the presence of an LPL. Our work, therefore, presents a detailed analysis of exploring the superconductivity mediated through a strong correlation in the presence of electromagnetic drive. Though our work explored the driven honeycomb lattice, but is not limited and is applicable to generic strongly correlated systems.

The authors thank Ying Su for helpful discussion at the initial stage of the work. Computer resources for numerical calculations were supported by the Institutional Computing Program at LANL. This work was carried out under the auspices of the U.S. DOE NNSA under contract No. 89233218CNA000001 through the LDRD Program. S.-Z. L was also supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, Condensed Matter Theory Program.