Tunning the tilt of the Dirac cone by atomic manipulations in 8Pmmn borophene

Abstract

Two dimensional quantum materials possessing Dirac cones in their spectrum are fascinating due to their emergent low-energy Dirac fermions. In 8Pmmn borophene the Dirac cone is furthermore tilted, which is a proxy for spacetime geometry, since the future light-cone depends on the underlying metric. Therefore it is important to understand the microscopic origin of the tilt. Here, based on ab-initio calculations, we decipher the atomistic mechanism of the formation of tilt. First, nearest-neighbor hop** on a buckled honeycomb lattice forms an upright Dirac cone. Then, the difference in the renormalized anisotropic second-neighbor hop**, formed by virtual hop**s on one-dimensional chains of atoms, tilts the Dirac cone. We construct an accurate tight-binding model on honeycomb graph for analytical investigation, and we find that substitution of certain boron atoms by carbon provides a way to change the tilt of the cone.

Introduction

The marriage between the mathematical concept of “topology” and quantum materials lead to the birth of “topological materials” and developments that followed afterwards. Can the (local) “geometry” play a similar role in combination with quantum materials? In the same way that the appearance of Dirac fermions in graphene and other Dirac materials can be attributed to an emergent Minkowski spacetime structure at distances much larger than the atomic distances, quantum materials with “tilted Dirac cone” can be naturally associated with an emergent spacetime (not merely the space) metric. In these class of materials the tilt is a proxy to spacetime geometry. We use Geometric Quantum Material (GQM) to refer to them.

Every periodic structure in quantum condensed matter is mounted on a mathematical object called lattice. Irrespective of which atoms one wishes to place on the sites of a given lattice, they come in 230 possible structures¹. The presence of lattice breaks translation and rotation, and some times parity and/or time reversal invariance of the vacuum². The lattice therefore breaks the Poincaré group³ thereby the connection between spin and statistics of the particles is lost and therefore one may have fermions with integer spin that is enforced by the irreducible representations of its space group (SG)⁴.

Is there any other interesting consequence that can be associated with the underlying SG? To set the stage for answering this question, let us start by a simple, but profound observation²: Consider a simple quantum mechanical hop** process on a lattice and think about the wave equation in the continuum limit. On the square lattice the Hamiltonian becomes − ℏ²∇²/(2m^*), while on the honeycomb lattice it becomes iℏv_Fσ. ∇ , namely the massless Dirac Hamiltonian⁵. Therefore despite that in the continuum limit, the lattice spacing is immaterial, but still remnants of the microscopic symmetries of the pertinent SG are manifested in the long-distance behavior and decide whether the structure of the ensuing spacetime is Galilean or Minkowskian. This can be regarded as an example of metamorphosis of atomic scale symmetry to a long-distance geometrical structure. The duality between atomic scale SG symmetry and long distance geometry is central to understand the geometric structure in GQMs.

What is the microscopic mechanism of such a duality between the microscopic structure and the long-distance (low-energy) characteristics in more complicated lattice structures? The long-distance behavior of interest for us is the tilted Dirac cone band whose Hamiltonian is ∝ (σ. ∇ ) + σ₀ζ. ∇ ^{6,7,8,9,10,11} where σ₀ is the 2 × 2 unit matrix. A vector-looking quantity ζ here determines the tilt of the Dirac cone. But at a much deeper level, it can be encoded into an emergent spacetime metric $d{s}^{2}=-{({v}_{{{{{{{{\rm{F}}}}}}}}}dt)}^{2}+{(d{{{{{{{\boldsymbol{x}}}}}}}}-{{{{{{{\boldsymbol{\zeta }}}}}}}}{v}_{{{{{{{{\rm{F}}}}}}}}}dt)}^{2}$^{9,10,12,13,14,15,16,17}, where v_F is the Fermi velocity. Therefore the tilt of a Dirac cone is actually a proxy for an emergent solid-state spacetime structure. Since the density of states is enhanced by a $1/\sqrt{1-{\zeta }^{2}}$, the tilted Dirac cone materials can generically give stronger responses to external stimuli. This enhancements nicely fits into spacetime description as a “redshift” factor¹⁸. Therefore it is crucial to understand the atomistic mechanism of the formation of the tilt in order to utilize it.

In this work we decipher the microscopic mechanism that determines the tilt parameter ζ. For this purpose we focus on the SG number 59 that describes the 8Pmmn structure of borophene^9,19,20. The tilted Dirac cone has been experimentally observed in the so called χ₃ structure of borophene²¹. The logic employed here is generic and is applicable to other SGs, and can even be used to discover further examples of GQM. We show that in this structure, the low-energy and high-energy degrees of freedom are nicely separated into two sublattices denoted by gray and teal circles in Fig. 1a, respectively. When the high energy sites available in this particular SG are integrated out, the underlying honeycomb lattice will be promoted to a “honeycomb graph”. Therefore the molecular orbitals play significant role in attaching new “graph edges” to the simple underlying honeycomb lattice. As we will see, this graph has the ability to lead to a controlable tilt and hence a tunable spacetime structure in the long distance. On such a graph, even if instead of fermions one places circuit elements, again a tilted Dirac cone can be obtained²². Therefore in addition to GQMs, the circuits can also emulate interesting spacetime structures with a larger degree of tunability.

Fig. 1: Electronic structure of pure 8 Pmmn borophene.

a Top and side view of crystal structure of 8 Pmmn borophene. The gray and faint teal circles denote inner (I) and ridge (R) boron atoms, respectively. b Density functional theory with Perdew-Burke-Ernzerhof parameterization (DFT-PBE) band structure of pristine 8 Pmmn borophene and (c) its three dimensional reconstruction. d First Brillouin zone and decomposition of elementary band representation the 8 Pmmn group. The ± are the eigenvalues of ${\tilde{C}}_{{{{{{{{\rm{2x}}}}}}}}}$, and ${\tilde{C}}_{{{{{{{{\rm{2y}}}}}}}}}$ operations at high-symmetry points. e The orbital-projected band structures for two atoms of 8 Pmmn borophene based on DFT-PBE. The Fermi energy (E_F) is set to the zero energy.

Results

Protection of the Dirac node

The relevant orbitals in the boron (as well as C) atom are 2p orbitals. The possibility of formation of sp² and sp³ hybridization establishes the honeycomb lattice²³, and structures such as 8Pmmn that involve buckled honeycomb networks as natural lattices for these atoms. As can be seen in Fig. 1a, there is a backbone (buckled) honeycomb sub-lattice denoted with gray circles that are called inner (I) sites. The rest of the lattice sites are called ridge (R) sites and are denoted by teal circles. First principle calculations indicate that the resulting Dirac cone is tilted^24,25 which is shown as red (low-energy) band in Fig. 1b and the three dimensional reconstruction of the band structure is shown in panel (c). The first thing that the 8Pmmn SG implies about the tilted Dirac cone is that on the MX and MY lines in the border of the Brillouin zone (BZ) of Fig. 1d, two bands “stick together” (as Kittel puts it²⁶) where the two-dimensional irreducible representations are protected by non-symmorphic elements²⁶. Then the compatibility relations gives the qualitative band picture of cat’s cradle shape²⁷ shown in panel (d). For pedagogical details of the derivation of this figure with group theory methods see the Supplementary Note 1, or more extended version of this paper^{2b depicts the BZ of original 8Pmmn lattice. The insight from the projected bands in the left and center columns in the second row of Fig. 1e is that both p_x and p_z orbitals of the R atoms are of comparable importance and must be incorporated into the atomic scale computation of the t₆₅ (dark green) and t₈₇ (light green). Similarly the first column of Fig. 1e suggests that the p_z orbitals of R and I atoms dominate in t₃₆ = t₃₅ hop** process. Table 1 shows the calculated values of these hop** parameters for pristine borophene (B₈) and C-doped borophene B₆C₂ using Wannier function29,30. The purpose of substituting C for B is to study its effect in the tilt of the Dirac cone. The substituted carbon dimers are placed in R and I positions, respectively. Using these parameters one can re-construct the ab initio bands with four p_z orbitals of four inner sites, and eight p_x and p_z orbitals of ridge sites. The nice coincidence of the original DFT bands with Wannier-interpolated bands in Supplementary Fig. 9 shows that the obtained t_ij values are reliable and the used atomic orbitals are adequate.}

Fig. 2: Coarse graining of 8Pmmn lattice into honeycomb graph.

a The lattice structure, unit cell, and hop** processes in the 8 Pmmn lattice. Inner sites (I) are denoted by teal circles while the ridge (R) sites are denoted by gray circles. b The Brillouin zone of 8 Pmmn lattice with two tilted Dirac cones. c The “effective” honeycomb graph of I atoms (low-energy degrees of freedom) obtained by elimination of R sites from the 8 Pmmn lattice. d two representations (rhombus and hexagonal) of the Brillouin zone of the effective honeycomb structure. The effective second/third neighbor hop**s (green/red) in (c) arise from the corresponding hop** path in (a) via the process of renormalization (see the section on Renormalization via molecular orbitals). As indicated in (d), the red (third neighbor) hop** shifts the Dirac cone, while the green (second neighbor) hop**s tilt the cone.

Table 1 Atomic and renormalized hop**s (in eV) for borophene and carbon-substituted borophene.

Although such an atomic picture might be satisfactory if one wishes to focus on the low-energy features of the tilted Dirac cone around the Fermi surface, but still working with a 12-band Hamiltonian is neither convenient, nor the essential long-range physics depends on so many short-distance details. To achieve an effective two-band model (for more details see Supplementary Note 2), we need to decimate the 8Pmmn lattice into an effective honeycomb graph shown in Fig. 2c where two possible ways of representing its BZ are depicted in Fig. 2d. On such a coarse-grained lattice, the virtual atomic hop** paths will be replaced by effective hop**s of the same color in panel (c). The connection between these effective hop**s and atomic hop**s t_ij is a nice example of renormalization that will be discussed now.

Renormalization via molecular orbitals

Anderson in his book maintains that renormalization is one of the pillars of condensed matter physics³¹. In this section we will show that the same concept is encoded into a local quantum chemistry of the 8Pmmn borophene. Since we are not interested in higher energy features that take place away from the tilted Dirac node, we build an effective picture based on Fig. 2c where the “effective” hop**s must be evaluated from the atomic scale data in the upper part of Table 1. The low-energy degrees of freedom are dominated by the p_z orbitals of the I sites. So we decimate the 8Pmmn lattice by elimination of the R sites. In doing so, the effective coarse-grained system becomes the honeycomb graph in Fig. 2c. The nearest neighbor hop**s denoted by black arrows take place within the low-energy subspace and are responsible for the formation of a parent Dirac cone from the p_z orbitals of the I sites. Slight anisotropy in the black arrows is known to shift the location of the Dirac cone^32,33 within the rhombus BZ of panel (d) that we ignore here. Green/red hop** processes are associated with the second/third neighbors of the honeycomb backbone that originate from the corresponding atomic process t_ij of panel (a) via the renormalization.

Let us see how the atomic processes t_ij in Fig. 2a are related to the effective hop** processes in Fig. 2c³⁴. For example consider the simplest third neighbor R sites 2 and 3 of adjacent unit cells in Fig. 2a that becomes possible via atomic hop**s t₂₇ and t₃₇ of Fig. 2a. The hop** process via the R site 7 is depicted in Fig. 3a. Assuming an on-site energy offset 2Δ for the R sites with respect to I site, an electron starting at the site 2 virtually hops to R site 7 and then returns to low-energy sector at site 3. Through this process it gains the following energy (see Fig. 3b)

$${t}^{{{{{{{{\rm{p}}}}}}}}}={{\Delta }}-\sqrt{2{t}_{27}^{2}+{{{\Delta }}}^{2}} \, < \, 0,$$

(1)

even in the limit of Δ → 0 this gives an energy lowering $-\sqrt{2}| {t}_{27}|$ that can be regarded as effective hop** between the third neighbor sites 2 and 3 of the I sublattice. To intuitively understand this formula, note that in the absence of the I site, t₂₇ = t₃₇ = 0, and hence both even and odd combinations of third neighbor atomic orbitals $\left\vert 2\right\rangle \pm \left\vert 3\right\rangle$ remain inert. But once the inner site is present, the atomic hop** t₂₇ causes a coupling of $\left\vert 7\right\rangle$ with the even-parity state $\left\vert 2\right\rangle +\left\vert 3\right\rangle$, leaving the odd parity state $\left\vert 2\right\rangle -\left\vert 3\right\rangle$ decoupled at zero energy. The coupling with the even parity combination provides a channel to lower the energy given by the effective hop** in Eq. (1). Furthermore, in the limit of Δ ≫ ∣t₂₇∣ the above formula reduces to the perturbatively appealing form $-2{t}_{27}^{2}/{{\Delta }}$. For pedagogical details of the above computation, please refer to Supplementary Note 3.

Fig. 3: Effective 3rd neighbor hop** t^p via the “step** stone” sites.

a Virtual hop** via the ridge site 7 generates an effective hop** t^p between 2 and its third neighbor site 3—i.e. site 3 in the adjacent unit cell in Fig. 2(a). b The odd parity combination of 2 and 3 remains decoupled, but the even parity combination hybridizes with ridge site to gain energy. 2Δ is difference of atomic on-site energy.

As a second example, consider the I site 3 in Fig. 2a and the same site in the unit cell above it (that we denote by ${3}^{{\prime} }$). These sites will be second neighbor on the coarse grained lattice and there are two hop** pathways $3\to 6\to 5\to {3}^{{\prime} }$ and $3\to 8\to 7\to {3}^{{\prime} }$ connecting them that are denoted by light and dark green dashed lines in Fig. 2a. The process of the calculation of the effective hop** amplitude $3\to {3}^{{\prime} }$ is similar for the above two paths. So we focus on the first one. In principle one must consider the energy gained by the lowest molecular orbitals formed by the above chain of sites. This has been done in Supplementary Note 3. The end result allows for a nice and intuitive interpretation depicted in Fig. 4(a–c): First, due to hop** t₆₅, the R sites 5 and 6 form bonding and anti-bonding molecular orbitals denoted by $\left\vert {\phi }_{1}\right\rangle$ and $\left\vert {\phi }_{3}\right\rangle$ in Fig. 4a. Then as depicted in panel (b), an electron gains energy by virtually hop** via the anti-bonding orbital $\left\vert {\phi }_{3}\right\rangle$ whose energy is now offset by t₆₅ + 2Δ. The hop**s connecting site 3 and ${3}^{{\prime} }$ to $\left\vert {\phi }_{3}\right\rangle$ are ${t}_{65}/\sqrt{2}$ where the $\sqrt{2}$ factor comes from the normalization $\left\vert {\phi }_{3}\right\rangle =(\left\vert {\phi }_{5}\right\rangle -\left.{\phi }_{6}\right\rangle )/\sqrt{2}$. In the final step as shown in Fig. 4c, the even-parity combination of $\left\vert 3\right\rangle$ and $\left\vert {3}^{{\prime} }\right\rangle$ is mixed with $\left\vert {\phi }_{3}\right\rangle$ to give an energy gain ${E}_{-}^{{{{{{{{\rm{even}}}}}}}}}=({t}_{56}/2+{{\Delta }})-\sqrt{{({t}_{56}/2+{{\Delta }})}^{2}+{t}_{63}^{2}}$. A similar contribution arises from the second path. Adding the two contributions we obtain the total renormalized hop** $\tilde{t}$ of Fig. 2c as

$$\tilde{t}= \frac{{t}_{56}+{t}_{87}+4{{\Delta }}}{2}\\ - \sqrt{{\left(\frac{{t}_{56}}{2}+{{\Delta }}\right)}^{2}+{t}_{63}^{2}}-\sqrt{{\left(\frac{{t}_{87}}{2}+{{\Delta }}\right)}^{2}+{t}_{73}^{2}}$$

(2)

in the above formula, the hop** parameters t₆₅ and t₈₇ are dominantly contributed by the p_x and p_z orbitals as the intensities of the p_x, p_z orbitals in the second row of Fig. 1e are dominant, while p_y orbital is faint. Similarly the two other renormalized hop** parameters t^x and $\bar{t}$ can be computed. For details please refer to Supplementary Information (SI). where we have shown that the elimination of higher energy states actually corresponds to the above simple molecular orbital analysis.

Fig. 4: Effective 2nd neighbor hop** $\tilde{t}$ via “step** stone” sites.

a Formation of bonding $\left\vert {\phi }_{1}\right\rangle$ and anti-bonding $\left\vert {\phi }_{2}\right\rangle$ orbitals between the ridge atoms 5 and 6. b Effective hop** $\tilde{t}$ between second neighbor inner sites 3 and ${3}^{{\prime} }$ can be formed by virtual hop** via the anti-bonding orbital $\left\vert {\phi }_{3}\right\rangle$ of two ridge atoms. 2Δ is difference of atomic on-site energy. E₀ is atomic on-site energy of $\left\vert 3\right\rangle$ and $\left\vert {3}^{{\prime} }\right\rangle$ and it is set to zero. c Energy diagram after the formation of molecular states which connect B3 to one of its successive third nearest neighbors through antibonding orbitals.

This is how the renormalized parameters of the coarse grained honeycomb graph model in the bottom part of Table 1 are computed from the ab initio data of the top part of the table. Note that if instead of 8Pmmn structure, we had a simple honeycomb lattice (such as in graphene), such a large values of second or third neighbor hop** given in Table 1 would be unthinkable as hop** between the atomic orbitals exponentially decays with distance. Therefore the virtual hop** via the ridge sites attaches a great importance to them as providers of channels for energy gain and ultimate formation of renormalized hop**s on a coarse grained lattice of inner sites. Furthermore, this is an example of how the molecular orbital play a significant role in the formation of longer range hop**s on the decimated 8 Pmmn lattice.

Effective coarse-grained model

Now that we have identified the p_z orbitals residing at the inner sites as the low-energy degrees of freedom, and have computed various renormalized hop**s between second and third neighbors, we are ready to write down a physically clear low-energy effective model that will straightforwardly demonstrate how the renormalized parameters ${t}^{{{{{{{{\rm{p}}}}}}}}},{t}^{{{{{{{{\rm{x}}}}}}}}},\tilde{t}$ and $\bar{t}$ can provide information about the position and the amount of the tilt.

The first neighbor hop**s denoted by black arrows in Fig. 2c are present even when there are no R sites. In the two-dimensional Hilbert space of A and B sublattice of such a honeycomb lattice, these hop**s contribute the usual off-diagonal term ${F}_{0}({{{{{{{\boldsymbol{k}}}}}}}})=-{\sum }_{\alpha = 1,2,3}{e}^{{{{{{{{\rm{i}}}}}}}}{{{{{{{\boldsymbol{k}}}}}}}}.{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{\alpha }}$ to the Bloch Hamiltonian, where the sum over α runs over three first neighbors in Fig. 2c. This is responsible for the formation of a pair of upright Dirac cones similar to graphene. Slight anisotropy of the honeycomb lattice of R sites will amount to a shift in the location of the Dirac cone³² which is irrelevant for our purposes. For concrete calculations, we assume a regular effective honeycomb lattice of bond length a for which the Dirac nodes in the rhombus BZ are at ${K}_{{{{{{{{\rm{b}}}}}}}}}^{\pm }=\frac{2\pi }{3a}(\hat{i}\pm \frac{1}{\sqrt{3}}\hat{j})$²³.

Next let us consider the third neighbor hop**s denoted by red arrows in Fig. 2c labeled by t^p (superscript “p” for pseudo, as they give rise to pseudogauge fields that shift the location of the Dirac node in opposite directions) and t^x (superscript “x” to emphasize the role of p_x orbitals of R sites). They contribute another off-diagonal term given by $={F}_{{{{{{{{\rm{xp}}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})={t}^{{{{{{{{\rm{p}}}}}}}}}{e}^{2{{{{{{{\rm{i}}}}}}}}a{k}_{x}}+2{t}^{{{{{{{{\rm{x}}}}}}}}}{e}^{-{{{{{{{\rm{i}}}}}}}}a{k}_{x}}\cos (\sqrt{3}a{k}_{y})$ to the Bloch Hamiltonian. Expanding this form factor around the Dirac nodes ${K}_{{{{{{{{\rm{b}}}}}}}}}^{\pm }$ above, gives rise to (i) a shift Δk_D = 2(t^p − t^x)/( ∓ 3at ± at^x) of the Dirac node, where t is the nearest neighbor hop** (assumed to be 1) and (ii) anisotropy in the Fermi velocity given by ${v}_{{{{{{{{\rm{F}}}}}}}}x}\to {v}_{{{{{{{{\rm{F}}}}}}}}x}\left(1+\frac{4{t}^{{{{{{{{\rm{p}}}}}}}}}+2{t}^{{{{{{{{\rm{x}}}}}}}}}}{3t}\right)$ and ${v}_{{{{{{{{\rm{F}}}}}}}}y}\to {v}_{{{{{{{{\rm{F}}}}}}}}y}\left(1-\frac{6{t}^{{{{{{{{\rm{x}}}}}}}}}}{3t}\right)$. Therefore the first and third neighbor hop**s establish the location of the (still upright) Dirac cone and determine its Fermi anisotropic velocities.

Now let us focus on the second neighbors (green arrows) in Fig. 2c that are the root cause of the tilt formation⁷. Since these hop**s are driven via virtual hop** through different arrangements of molecular orbitals, there are two types of them denoted by solid ($\bar{t}$) and dashed ($\tilde{t}$) green lines. These hop**s being second neighbor, connect two atoms on the same sublattice, and therefore contribute to the diagonal terms, namely AA and BB components of the effective 2 × 2 Hamiltonian matrix. Among the matrices σ_μ with μ = 0…3, only σ₀ and σ_z can contribute diagonal terms. So now one has to decide whether these diagonal terms come with the same sign (σ₀ term → tilt) or opposite signs (σ_z term → gap). There are two ways to see that this term must be proportional to the σ₀: (i) Analysis of the irreducible representations and compatibility relations of the original 8Pmmn structure in Fig. 1d shows that the crossing of the red bands is protected by the glide elements of the 8 Pmmn lattice (see Supplementary Note 1 for details). Since the effective theory has to obey this protection against gap opening, the σ_z term is ruled out. (ii) Consider the renormalized lattice itself and focus on the solid green line in Fig. 2c. If the hop** between 1 and 3 in Fig. 1a contributes to AA term, the hop** between 2 and 4 contribute to BB term. Both these contributions arise from the p_z orbitals of these atoms via intermediate hop** through p_z orbital of atom 5. Apparently for p_z orbitals the “northwest” (1 → 5 → 3) and “northeast" (4 → 5 → 2) hop**s are identical. Similar arguments holds for the dashed green line in Fig. 2c. In this case AA (BB) term is generated by 3 → 6 → 5 → 3 (2 → 6 → 5 → 2) path. Again the p_x orbitals of the 5, 6 atoms symmetrically connect $3\to {3}^{{\prime} }$ and $2\to {2}^{{\prime} }$ (remember ${3}^{{\prime} }$ is the same as 3 but in adjacent unit cell. Similarly for ${2}^{{\prime} }$.), thereby giving identical AA and BB terms in the effective Hamiltonian.

Therefore the effective Hamiltonian becomes (for more details see Supplementary Note 2)

$$\begin{array}{l}{H}_{{{{{{{{\rm{eff}}}}}}}}}({{{{{{{\bf{k}}}}}}}})=\left(\begin{array}{ll}{f}_{{{{{{{{\rm{tilt}}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})&f({{{{{{{\boldsymbol{k}}}}}}}})\\ {f}^{* }({{{{{{{\boldsymbol{k}}}}}}}})&{f}_{{{{{{{{\rm{tilt}}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})\end{array}\right),\,f({{{{{{{\boldsymbol{k}}}}}}}})={F}_{{{{{{{{\rm{xp}}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})+{F}_{0}({{{{{{{\boldsymbol{k}}}}}}}}),\\ {f}_{{{{{{{{\rm{tilt}}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})=2\tilde{t}\cos (\sqrt{3}a{k}_{y})+4\bar{t}\cos (3a{k}_{x}/2)\cos (\sqrt{3}a{k}_{y}/2).\end{array}$$

(3)

Taylor expanding the diagonal f_tilt term around the Dirac node formed by off-diagonal terms gives,

$${\zeta }_{{{{{{{{\rm{x}}}}}}}}}=0,\,\,{\zeta }_{{{{{{{{\rm{y}}}}}}}}}=\pm 2\frac{\tilde{t}-\bar{t}}{t},$$

(4)

where ± corresponds to the valley around which the expansion is performed.

The above equation indicates that the tilt arises from the difference of the second neighbor hop**s $\tilde{t}$ and $\bar{t}$ that in turn are generated via the p_x and p_z orbitals of the R atoms. An immediate suggestion of the above model is to (partially) replace the R-site boron atoms by carbon atoms to see whether the tilt is changed or not. In Fig. 5(a–j) we have replaced two of the boron atoms with C atoms. For this purpose there are two choices: (i) To place a carbon dimer on the R sites as in the panel (a) or (ii) to place the carbon atoms in the I sites as in panel (f). The results are summarized in Table 1. For case (i), the location of the Dirac node is shifted towards the Γ point and its tilt increases. Placing carbon atoms in the R sites shifts the 5 and 6 sites to higher energies, thereby generating larger $\tilde{t}$ and $\bar{t}$ (see Table 1) that ultimately increases the tilt from the ζ_y = 0.46 of the pristine borophene to ζ_y = 0.59 in B₆C₂-R-[C5&C6]. Our picture provides also a way to decrease the tilt parameter. In this case, the hybridization of the p_z orbitals of sites 2, 3 with the R sites (5, 6) reduces as in Table 1 and hence the resulting $\bar{t}$ and likewise $\tilde{t}$ are scaled down, thereby reducing the tilt to 0.36 in B₆C₂-I-[C2&C3]. The above values of the tilt ζ_y are calculated based on Eq. (4) and as detailed in Supplementary Note 4, are in good agreement with the corresponding values directly extracted from DFT bands.

Fig. 5: Manipulation of the tilt of Dirac cone by carbon substitution in 8 Pmmn borophene.

a Top view of crystal structure of B₆C₂-R-[C5&C6] and b its Brillouin zone. The red circles denote carbon atoms. c Density functional theory with Perdew-Burke-Ernzerhof (DFT-PBE) band structure of B₆C₂-R-[C5&C6], d its orbital-projected band structures for two atoms and (e) its three dimensional reconstruction. f–j The same as (a–e) for B₆C₂-I-[C2&C3]. The horizontal arrows in panels c and h indicate the direction of the displacement of the Dirac node with respect to the parent B₈. The vertical arrows in a and f show the direction of movement of the lattice sites upon carbon substitution.

As shown in Supplementary Fig. 6 of Supplementary Note 4, for do** of two carbon atoms into the structural unit of B₈, the R-site configurations (c) and (k) have the lower energy than all other configurations, with staggered configuration (k) having slightly lower energy than (c). For a perfect (translationally invariant) do** of two C into the 8 Pmmn borophene structure, the changes in the tilt are discrete. However, the concentration of C atoms can be continuously varied. Statistical averaging for a concentration x that continuously varies between 0 and 2/8 is expected to generate a continuously varying tilt. The dominant configurations will be the R-site dimers and staggered R-site configurations of panel (c) and (k) of Supplementary Fig. 6. The lack of perfect backscattering of Dirac electrons is expected to protect the Dirac cone against the Anderson localization.

Conclusions

Based on our ab initio calculations, we have identified the p_z orbitals of the I sites as real space sublattice on which the low-energy degrees of freedom in 8 Pmmn lattice reside. The effective hop**s between these sites are obtained via renormalization that encodes the virtual hop** via R-sites (Supplementary Note 5). This gives a physically clear picture of the formation of the tilt in two-dimensional Dirac cone of 8 Pmmn borophene. In our picture, tilt arises from a competition between two renormalized hop**s $\tilde{t}$ and $\bar{t}$ between the second neighbor I-sites. The former involves virtual hop** via two R sites (and 2p_x orbitals of the R sites), while the latter involves one R site. This builds in a natural difference between $\bar{t}$ and $\tilde{t}$. That is why the pristine borophene has a substantial tilting. Within this picture it is natural to expect that replacing the ridge (inner) atoms by C increase (decreases) the tilt. The rest of the effective hop**s determine the location of the (protected) Dirac node and the anisotropy in the Fermi velocity.

The logic of our work can be applied to other SGs to discover more GQM within a class of 2D materials that afford to provide a “parent” upright Dirac cone: Molecular orbitals in certain SGs can mediate longer range hop**s on a backbone lattice of Dirac fermions by promoting it to appropriate graph. The resulting graph in turn deforms the Minkowski spacetime of the Dirac fermions into a metric involving spatio-temporal elements. The ability to tune the tilt parameter is tantamount to controllability of the ensuing solid-state spacetime structure that emerges at long distances. Dirac materials subject to “gravitational” (i.e. geometric) disorder^35,36 find their salient materialization in the present context when the carbon is randomly substituted for boron. Assuming a do** fraction x of carbon atoms that can be continuously tuned, the most favorable configurations for the C atoms are R-site dimers and staggered R-site (Supplementary Fig. 6c, and k, respectively). Random substitutions of this form are expected to lead to continuous variation of tilt as a function of x. The absence of backscattering for Dirac fermions protects it against Anderson localization³⁷.

The relation between certain graphs and their continuum limit as space geometries is well known^38,39,40. In the present context, each edge of the graph is associated with a hop** energy (frequency) scale. This is how space and time are mingled and a spacetime geometry (rathern than space geometry) emerges at long distances. This is how a solid-state platform can promote a simple lattice that supports Dirac fermions into a rich graph where vierbeins can be attached to the Dirac fermions^41,42. This enriches the physics of Dirac materials and promotes them to GQMs where a variety of spacetime geometries can be fabricated that might not even have any analogue in the cosmos. On the 2D materials side, there will be a plenty of room to explore the effects of “geometric” forces or even synthesise non-Abelian gauge fields⁴³ that are likely to lead to better control in electronic/optical devices, as the effects of spacetime curvature^44,45 can be much stronger GQMs. Furthermore, the geometry of our emergent spacetime in GQMs roots in the Coulomb forces that are much stronger than the gravitational forces and unlike the gravity, the Coulomb forces can be both attractive and repulsive. Therefore the geometric structure of GQMs seems to be much richer than the Einsten’s gravity. In three dimensional tilted Dirac/Weyl fermions the resulting geometric theory can be even more interesting: The fermions in such materials are chiral and hence a chiral geometric theory can be relevant to GQM. The developments in GQMs will allow us to emulate aspects of spacetime geometry that is not easily accessible in the cosmos. GQMs have a potential to equip Dirac fermions with types of vielbeins⁴¹ that are beyond the spatial veilbeins of the strain/dislocation paradigm⁴².

Methods

For DFT calculation we have used pseudopotential Quantum Espresso code based on plane wave basis set within the generalized gradient approximation in the Perdew-Burke-Ernzerhof (PBE) parameterization⁴⁶. Simulation of borophene rectangular unit cells is based on the slab model having a 25 Å vacuum separating slabs. We also consider monolayer of 8Pmmn structure, where some of the B atoms are substituted by C atoms in the form of B_8−xC_x (x = 0, 1, 2). The obtained structural properties after the ionic relaxations such as lattice parameters, x, y, and z component of the B and C atoms in crystal coordinates are shown in Supplementary Table II of SI for pure borophene (B₈) and C-doped systems B_8−xC_x. The uniform k-point grids of 24 × 24 × 1 are used for the self-consistent field calculations of all systems. The Kinetic energy cut-offs for the wavefunctions and the charge density are 850 and 8500 eV, respectively. For each systems, the Broyden-Fletcher-Goldfarb-Shanno quasi-Newton algorithm is used to relax the internal coordinates of the B and C atoms and possible distortions with convergence threshold on forces for ionic minimization as small as 10⁻⁴ eVÅ⁻¹. Also the generation of new edges on the honeycomb graph is based on the standard molecular orbital theory that can also be formulated in the language of the renormalization. The protection of the Dirac node can be directly seen by looking into irreducible representation of the little groups. The full pedagogical details of the above procedure is presented in Supplementary Notes 4 and 5 of the SI.