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 hopping 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 iℏvFσ. ∇ , 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 vF 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 borophene9,19,20. 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.
figure 1

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 (EF) is set to the zero energy.

Full size image

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 tilted24,25 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 paper28.

Of course the protection of the tilted Dirac cone by the underlying SG is interesting, but it is not essential for the main purpose of this paper, namely the “tilt” of the Dirac cone. The symmetry considerations do not explain why and how the tilt parameter ζ is formed. In order to “manipulate” the tilt of the Dirac cone at will, one needs a microscopically detailed understanding of the root cause of formation of tilt in the Dirac cone. To achieve this, we need to identify the low-energy degrees of freedom that give rise to the tilted Dirac cone dispersion. For this purpose in Fig. 1e we have plotted an orbital projected representation of the band structure that resolves the contribution of atoms 2 (I) and 5 (R). As can be seen, the dominant contributions to the tilted Dirac dispersion comes from the pz orbitals of the I atoms (top left plot in panel (e)). The R atoms also contribute via their pz and px orbitals (bottom left and center). The remaining two boxes on the top row indicate little contributions from the px and py orbitals of the I atoms that arises from the buckling of the honeycomb sublattice.

In Fig. 2a we have labeled atomic hoppings paths with green and red dashed lines that connect I sites by virtual hoppings through R sites. There is only first neighbor direct hopping between the I sites denoted by black arrows. The appropriate atomic orbitals involved in forming the above microscopic tij hoppings can be extracted from the density functional theory (DFT) calculation. Working in a gauge that all tij’s are real, the hermiticity implies tij = tji. Fig. 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 px and pz 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 pz orbitals of R and I atoms dominate in t36 = t35 hopping process. Table 1 shows the calculated values of these hopping 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 pz orbitals of four inner sites, and eight px and pz orbitals of ridge sites. The nice coincidence of the original DFT bands with Wannier-interpolated bands in Supplementary Fig. 9 shows that the obtained tij values are reliable and the used atomic orbitals are adequate.

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

a The lattice structure, unit cell, and hopping 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 hoppings (green/red) in (c) arise from the corresponding hopping path in (a) via the process of renormalization (see the section on Renormalization via molecular orbitals). As indicated in (d), the red (third neighbor) hopping shifts the Dirac cone, while the green (second neighbor) hoppings tilt the cone.

Full size image
Table 1 Atomic and renormalized hoppings (in eV) for borophene and carbon-substituted borophene.
Full size table

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 hopping paths will be replaced by effective hoppings of the same color in panel (c). The connection between these effective hoppings and atomic hoppings tij 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” hoppings 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 pz 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 hoppings denoted by black arrows take place within the low-energy subspace and are responsible for the formation of a parent Dirac cone from the pz orbitals of the I sites. Slight anisotropy in the black arrows is known to shift the location of the Dirac cone32,33 within the rhombus BZ of panel (d) that we ignore here. Green/red hopping processes are associated with the second/third neighbors of the honeycomb backbone that originate from the corresponding atomic process tij of panel (a) via the renormalization.

Let us see how the atomic processes tij in Fig. 2a are related to the effective hopping 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 hoppings t27 and t37 of Fig. 2a. The hopping 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 hopping 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 hopping 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 hopping 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 hopping tp via the “stepping stone” sites.
figure 3

a Virtual hopping via the ridge site 7 generates an effective hopping tp 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.

Full size image

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 hopping 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 hopping 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 hopping 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 hopping via the anti-bonding orbital \(\left\vert {\phi }_{3}\right\rangle\) whose energy is now offset by t65 + 2Δ. The hoppings 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 hopping \(\tilde{t}\) 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 hopping parameters t65 and t87 are dominantly contributed by the px and pz orbitals as the intensities of the px, pz orbitals in the second row of Fig. 1e are dominant, while py orbital is faint. Similarly the two other renormalized hopping parameters tx 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 hopping \(\tilde{t}\) via “stepping stone” sites.
figure 4

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 hopping \(\tilde{t}\) between second neighbor inner sites 3 and \({3}^{{\prime} }\) can be formed by virtual hopping 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.

Full size image

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 hopping given in Table 1 would be unthinkable as hopping between the atomic orbitals exponentially decays with distance. Therefore the virtual hopping via the ridge sites attaches a great importance to them as providers of channels for energy gain and ultimate formation of renormalized hoppings 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 hoppings on the decimated 8 Pmmn lattice.

Effective coarse-grained model

Now that we have identified the pz orbitals residing at the inner sites as the low-energy degrees of freedom, and have computed various renormalized hoppings 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 hoppings 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 hoppings 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 hoppings denoted by red arrows in Fig. 2c labeled by tp (superscript “p” for pseudo, as they give rise to pseudogauge fields that shift the location of the Dirac node in opposite directions) and tx (superscript “x” to emphasize the role of px 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 ΔkD = 2(tp − tx)/( ∓ 3at ± atx) of the Dirac node, where t is the nearest neighbor hopping (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 hoppings 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 hoppings are driven via virtual hopping through different arrangements of molecular orbitals, there are two types of them denoted by solid (\(\bar{t}\)) and dashed (\(\tilde{t}\)) green lines. These hoppings 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 σμ with μ = 0…3, only σ0 and σz 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 (σz 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 σz term is ruled out. (ii) Consider the renormalized lattice itself and focus on the solid green line in Fig. 2c. If the hopping between 1 and 3 in Fig. 1a contributes to AA term, the hopping between 2 and 4 contribute to BB term. Both these contributions arise from the pz orbitals of these atoms via intermediate hopping through pz orbital of atom 5. Apparently for pz orbitals the “northwest” (1 → 5 → 3) and “northeast" (4 → 5 → 2) hoppings 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 px 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 ftilt 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 hoppings \(\tilde{t}\) and \(\bar{t}\) that in turn are generated via the px and pz 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 ζy = 0.46 of the pristine borophene to ζy = 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 pz 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 ζy 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.
figure 5

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.

Full size image

As shown in Supplementary Fig. 6 of Supplementary Note 4, for doping 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) doping 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 pz orbitals of the I sites as real space sublattice on which the low-energy degrees of freedom in 8 Pmmn lattice reside. The effective hoppings between these sites are obtained via renormalization that encodes the virtual hopping 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 hoppings \(\tilde{t}\) and \(\bar{t}\) between the second neighbor I-sites. The former involves virtual hopping via two R sites (and 2px 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 hoppings 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 hoppings 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) disorder35,36 find their salient materialization in the present context when the carbon is randomly substituted for boron. Assuming a doping 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 known38,39,40. In the present context, each edge of the graph is associated with a hopping 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 fermions41,42. 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 curvature44,45 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 B8−xCx (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 B8−xCx. 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.