Zero Modes and Global Antiferromagnetism in Strained Graphene

A novel magnetic ground state is reported for the Hubbard Hamiltonian in strained graphene. When the chemical potential lies close to the Dirac point, the ground state exhibits locally both the Néel and ferromagnetic orders, even for weak Hubbard interaction. Whereas the Néel order parameter remains of the same sign in the entire system, the magnetization at the boundary takes the opposite sign from the bulk. The total magnetization vanishes this way, and the magnetic ground state is globally only an antiferromagnet. This peculiar ordering stems from the nature of the strain-induced single-particle zero-energy states, which have support on one sublattice of the honeycomb lattice in the bulk, and on the other sublattice near the boundary of a finite system. We support our claim with the self-consistent numerical calculation of the order parameters, as well as by the Monte Carlo simulations of the Hubbard model in both uniformly and nonuniformly strained honeycomb lattice. The present result is contrasted with the magnetic ground state of the same Hubbard model in the presence of a true magnetic field (and for vanishing Zeeman coupling), which is exclusively Néel ordered, with zero local magnetization everywhere in the system.

Popular Summary

One of graphene’s most appealing features is the interplay between the membrane’s geometry and its electronic structure. The effect of strain in graphene is felt by the electrons as an effective magnetic field. For deep mathematical reasons that are collectively known as “index theorems,” this effective magnetic field leads to the formation of a flat zero-energy band, which in graphene is half filled with electrons. In general, in the presence of electron-electron Coulomb interactions, flat bands have long been known to lead to magnetism. We examine the magnetic ground state in graphene under strain and show that, surprisingly, the state can only locally be characterized as a ferromagnet; its total magnetization is actually zero. In contrast, the global staggered magnetization is finite. The ground state of strained graphene can therefore be called a “global antiferromagnet.”

We study the Hubbard model, in which electrons interact only when they occupy the same site, with the strain implemented as a particular modulation of how the electrons hop between atoms. We allow both for uniform and nonuniform strain-induced effective magnetic fields and show that the zero-energy states that reside in the bulk of the sample and at the edge discriminate between the two sublattices of the honeycomb lattice. This finding is a consequence of the index theorem, which connects the degeneracy of the zero-energy Hilbert subspace to the total magnetic flux. In the presence of the Hubbard repulsion, however, this concomitant sublattice structure has an interesting consequence: In its attempt to be both an antiferromagnet and a local ferromagnet, the local magnetization in the ground state has to switch sign when going from the bulk to the edge. We show, using numerical analyses and Monte Carlo calculations, that the integrated magnetization vanishes, whereas the global magnetization is finite.

Experimental verification of this novel form of magnetism is a nontrivial challenge. Such an experimental feat would further our understanding of basic electronic behavior and open new possibilities for applications of strongly correlated materials.

Figure 1

A realization of strained honeycomb lattice that yields a finite axial magnetic field. Colors at each site correspond to the parameter $\chi$ in Eq. (3), which defines the modification of the hopping amplitude along each bond.

Figure 2

Top: Spatial distribution of the symmetric combination $(|S⟩)$ of the wave functions $|\pm \delta ⟩$, living on the $A$ sublattice, in a system of 294 sites or $r_{\text{max}}=7$. Bottom: Spatial distribution of their antisymmetric combination $(|AS⟩)$, living on the $B$ sublattice, in the same system. The axial magnetic field is set to be uniform here; i.e., $\chi (\mathcal{R})=q{\mathcal{R}}^2$, with $q=0.02$.

Figure 3

Upper panels: Spatial variation of the local Néel order parameter ($N$), defined in Eq. (9), obtained from the self-consistent solution of the magnetic ground state, for various subcritical on-site Hubbard interaction and roughly uniform axial magnetic fields. The strength of the Hubbard interaction in each plot reads as $U=0.1$ (red line), 0.2 (black line), 0.3 (green line), 0.4(blue line), 0.5 (magenta line), 0.6 (orange line), 0.7(cyan line) from bottom to top. Strength of the uniform axial field reads as $b=0.025b_0$, $0.035b_0$, $0.045b_0$ from left to right, where $b_0=\hslash /(e{a}^2)\sim 10^4$ T is the axial field associated with the lattice spacing $a\approx 2.5\AA$. Lower panels: Spatial variation of local ferromagnetic order parameter ($M$) obtained from the same self-consistent solution for the magnetic ground state. Here, $N$ is the average Néel order parameter per unit cell, and $M$ is the total magnetization, in a quasicircular ring.

Figure 4

Upper panels: Spatial variation of the local Néel ($N$) order parameter, obtained from the self-consistent solution of the magnetic ground state, as in Eq. (9), for various subcritical Hubbard interactions, and bell-shaped localized axial flux peaked around the center of the system. The strength of the Hubbard interaction in each plot reads as $U=0.1$ (red line), 0.2 (black line), 0.4 (blue line), 0.6 (orange line), 0.7 (dark green line) from bottom to top. Total flux of the localized axial field reads as ${\mathrm{\Phi }}_{\text{total}}=7.85{\mathrm{\Phi }}_0$, $9.42{\mathrm{\Phi }}_0$, $10.99{\mathrm{\Phi }}_0$ from left to right, where ${\mathrm{\Phi }}_0$ is the flux quanta. Inset in upper panels: Variation of $N(r)$ at the center of the system with the strength of the on-site interaction. Lower panels: Spatial variation of local ferromagnet order ($M$) in the same magnetic ground state for $U=0.1$ (red line), 0.2 (black line), 0.4 (dark green line), 0.6 (blue line), 0.7 (magenta line). Total flux of the localized axial field in the lower panel is same as in upper one. Here, $N$ and $M$ are the same as in Fig. 3.

Figure 5

Left: Spatial variation of the Néel (solid lines, with $x=N$) and ferromagnetic (black dots, with $x=M$) order parameters per unit cell, in the presence of uniform true magnetic field of strength 250 T. Right: The same quantities in a localized true magnetic field, of the strength 650 T at the center of the system. The average magnetic field at each quasicircular ring of the honeycomb lattice decreases as $B(r)=(690-40r)$ T, towards the boundary of the system. Strength of the Hubbard interaction reads as $U=0.4$ (red line), 0.5 (blue line), 0.6 (dark green line), 0.7 (purple line). The ferromagnetic order is identically zero for all values of $U$, both in uniform and nonuniform true magnetic fields.

Figure 6

Quantum Monte Carlo simulations on a 600-site flake. The top panels plot the spin-spin correlation functions between the reference point ${\mathit{i}}_R=(-1/2,-1/2\sqrt{3})$ and other sites of the lattice for both the uniform $\chi (\mathcal{R})=q{\mathcal{R}}^2$ and local $\chi (\mathcal{R})=q\log (\mathcal{R})$ axial fields. The two bottom panels plot the integrated spin-spin correlations $S(R)$, defined in Eq. (12), exhibiting, at finite values of $q$, edge-compensated antiferromagnetic spin structure.

Figure 7

Quantum Monte Carlo simulations on a 600-site flake. Top panels: Spin-spin correlation functions between the reference point ${\mathit{i}}_{\mathrm{R}}=(-1/2,-1/2\sqrt{3})$ and other sites of the lattice in the absence of axial field [$\chi (\mathcal{R})=0$] (left), and in the presence of uniform [$\chi (\mathcal{R})=q{\mathcal{R}}^2$] (middle) and local [$\chi (\mathcal{R})=q\log (\mathcal{R})$] (right) axial fields. Bottom panels: Integrated spin-spin correlations $S(R)$ [see Eq. (12)] at $U/t=4$ for the uniform $\chi (\mathcal{R})=q{\mathcal{R}}^2$ (left) and local $\chi (\mathcal{R})=q\log (\mathcal{R})$ (right) axial fields.