Non-Euclidean geometry, nontrivial topology and quantum vacuum effects

Space out of a topological defect of the Abrikosov-Nielsen-Olesen vortex type is locally flat but non-Euclidean. If a spinor field is quantized in such a space, then a variety of quantum effects is induced in the vacuum. Basing on the continuum model for long-wavelength electronic excitations, originating in the tight-binding approximation for the nearest neighbor interaction of atoms in the crystal lattice, we consider quantum ground state effects in monolayer structures warped into nanocones by a disclination; the nonzero size of the disclination is taken into account, and a boundary condition at the edge of the disclination is chosen to ensure self-adjointness of the Dirac-Weyl Hamiltonian operator. In the case of carbon nanocones, we find circumstances when the quantum ground state effects are independent of the boundary parameter and the disclination size.


Introduction
Usually, the effects of non-Euclidean geometry are identified with the effects which are due to the curvature of space. It can be immediately shown that this is not the case and there are spaces which are flat but non-Euclidean; moreover, such spaces are of nontrivial topology.
A simplest example is given by a two-dimensional space (surface) which is obtained from a plane by cutting a segment of a certain angular size and then sewing together the edges. The resulting surface is the conical one which is flat but has a singular point corresponding to the apex of the cone. To be more precise, the intrinsic (Gauss) curvature of the conical surface is proportional to the two-dimensional delta-function placed at the apex; the coefficient of proportionality is the deficit angle. Topology of the conical surface with a deleted apex is nontrivial: π 1 = Z, where π 1 is the first homotopy group and Z is the set of integer numbers. Usual cones correspond to positive values of the deficit angle, i.e. to the situation when a segment is deleted from the plane. But one can imagine a situation when a segment is added to the plane; then the deficit angle is negative, and the resulting flat surface can be denoted as a saddle-like cone. The deleted segment is bounded by the value of 2π, whereas the added segment is unbounded. Thus, deficit angles for possible conical surfaces range from −∞ to 2π.
It is evident that an apex of the conical surface with the positive deficit angle can play a role of the convex lens, whereas an apex of the conical surface with the negative deficit angle can play a role of the concave lens. Really, two parallel trajectories coming from infinity towards the apex from different sides of it, after bypassing it, converge (and intersect) in the case of the positive deficit angle, and diverge in the case of the negative deficit angle. This demonstrates the non-Euclidean nature of conical surfaces, providing a basis for understanding such physical objects as cosmic strings.
Conical space emerges inevitably as an outer space of any topological defect in the form of the Abrikosov-Nielsen-Olesen (ANO) vortex [1,2]. Topological defects are produced as a consequence of phase transitions with spontaneous breakdown of continuous symmetry in various physical systems, in particular, in superfluids, superconductors and liquid crystals. Cosmic strings with a specific gravitational lensing effect (doubling the image of an astrophysical object) are the ANO vortices which are produced as a result of phase transitions at the early stage of evolution of the Universe, see reviews in Refs. [3,4]. Otherwise, in micro-and nanophysics, a wealth of new phenomena, suggesting possible applications to technology and industry, is promised by a synthesis in this century of strictly two-dimensional atomic crystals (for instance, a monolayer of carbon atoms, graphene, [5,6]). Topological defects (disclinations) on such layers are similar to the transverse sections of cosmic strings. A disclination warps a sheet of a layer, rolling it into a nanocone; moreover, a physically meaningful range of values of the deficit angle is extended to include also negative values which correspond to saddle-like cones or cosmic strings with negative tension.
While considering the effect of the ANO vortex on the vacuum of quantum matter, the following circumstance should be taken into account: since the vacuum of quantum matter exists outside the ANO vortex core, an issue of the choice of boundary conditions at the edge of the core is of primary importance. The most general boundary condition for the matter wave function at the core edge is given by requiring self-adjointness of the Hamiltonian operator (energy operator in firstquantized theory).
In the present paper we consider quantum effects which are induced in the vacuum of the second-quantized pseudorelativistic gapless (i.e. massless) spinor field in (2+1)-dimensional space-time which is a section orthogonal to the ANO vortex axis; hence the Hamiltonian operator takes form H = −iα·∇, where covariant derivative ∇ includes both the affine and bundle connections (natural units = v F = 1 are used, with the Fermi velocity, v F , becoming the velocity of light, c, in the truly relativistic case). Condensed matter systems with such a behavior of low-energy electronic excitations are known as the two-dimensional Dirac materials comprising a diverse set ranging from honeycomb crystalline structures (graphene [5], silicene and germanene [7]) to high-temperature d-wave superconductors, superfluid phases of helium-3 and topological insulators, see review in [8]. We focus on the quantum ground-state effects (induced R-current and pseudomagnetic field) of electronic excitations in graphitic nanocones, although our consideration is quite general to also be relevant for nanocones of the nongraphitic nature as well; the finite size of a disclination at the conical apex is taken into account.

Continuum model description of electronic excitations in monolayer atomic crystals with a disclination
The squared length element of the conical surface is where ν = (1 − η) −1 and 2πη is the deficit angle. In the case of cosmic strings, the present-day astrophysical observations restrict the values of parameter η to range 0 < η < 10 −6 (see, e.g., [9]). A natural way of producing local curvature in the honeycomb lattice of graphene, silicene or germanene is by substitung some of hexagons by pentagons (positive curvature) and heptagons (negative curvature). Thus, in the case of crystalline nanocones, parameter η takes discrete values: η = N d /6, where N d is an integer which is smaller than 6. A disclination in the honeycomb lattice results from a substitution of a hexagon by a polygon with 6 − N d sides; polygons with N d > 0 (N d < 0) induce locally positive (negative) curvature, whereas the crystalline sheet is flat away from the disclination, as is the conical surface away from the apex. In the case of nanocones with N d > 0, the value of N d is related to apex angle δ, sin δ 2 = 1 − N d 6 , and N d counts the number of sectors of the value of π/3 which are removed from the crystalline sheet. If N d < 0, then −N d counts the number of such sectors which are inserted into the crystalline sheet. Certainly, polygonal defects with N d > 1 and N d < −1 are mathematical abstractions, as are cones with a pointlike apex. In reality, the defects are smoothed, and N d > 0 counts the number of the pentagonal defects which are tightly clustered producing a conical shape; graphitic nanocones with the apex angles δ = 112.9 • , 83.6 • , 60.0 • , 38.9 • , 19.2 • , which correspond to the values N d = 1, 2, 3, 4, 5, were observed experimentally, see [10] and references therein. Theory also predicts an infinite series of the saddle-like nanocones with quantity −N d counting the number of the heptagonal defects which are clustered in their central regions. Saddle-like nanocones serve as an element which is necessary for joining parts of carbon nanotubes of differing radii. On the basis of the long-wavelength continuum model originating in the tight-binding approximation for the nearest-neighbor interactions in the honeycomb crystalline lattice, it was proved [11] that the bundle connection effectively appears in addition to the affine connection of the nanocone, and the Hamiltonian operator takes form and matrix R exchanges the sublattice indices, as well as the valley indices, and commutes with H (2). Note that in the case of a cosmic string quantity Φ is the flux of a gauge vector field corresponding to the generator of a spontaneously broken continuous symmetry. Both R and α-matrices can be chosen in the block-diagonal form, (σ j with j = 1, 2, 3 are the Pauli matrices). The solution to the stationary Dirac- where the radial function satisfy the system of first-order differential equations Let us consider nanocones with N d = 1, 2, 3, 4, 5 (1 < ν < 7), as well as with , and introduce positive quantity which exceeds 1 at N d = 3, 4, 5 (2 ≤ ν < 7) only; here sgn(u) is the sign function.
From the whole variety of quantum effects in the ground state of electronic excitations (see [11,12,13]), our focus will be on the induced specific current (R-current) which is defined by expression The pseudomagnetic field strength, B I (x), is also induced in the ground state, as a consequence of the analogue of the Maxwell equation, Using (4) and (5), one gets j r = 0, and the only component of the induced ground state current, is independent of the angular variable. The induced ground state field strength is also independent of the angular variable, being directed orthogonally to the conical surface, with the total flux where it is assumed without a loss of generality that a nanocone is of a rotationally invariant shape with r max being its radius and r 0 being the radius of a disclination, r max ≫ r 0 .

Self-adjointness and choice of boundary conditions
Let us note first, that (2) is not enough to define the Hamiltonian operator rigorously in a mathematical sense. To define an operator in a unambiguous way, one has to specify its domain of definition. Let the set of functions ψ be the domain of definition of operator H, and the set of functionsψ be the domain of definition of its adjoint, operator H † . Then the operator is Hermitian (or symmetric in mathematical parlance), It is evident that condition (14) can be satisfied by imposing different boundary conditions for ψ andψ. But, a nontrivial task is to find a possibility that a boundary condition forψ is the same as that for ψ; then the domain of definition of H † coincides with that of H, and operator H is self-adjoint (for a review of the Weylvon Neumann theory of self-adjoint operators see [14,15]). The action of a selfadjoint operator results in functions belonging to its domain of definition only, and a multiple action and functions of such an operator, for instance, the resolvent and evolution operators, can be consistently defined. Thus, in the case of a surface of radius r max with a deleted central disc of radius r 0 , we have to ensure the validity of relationsψ † α r ψ meaning that the quantum matter excitations do not penetrate outside. It is implied that functions ψ andψ are differentiable and square-integrable. As r max → ∞, they conventionally turn into differentiable functions corresponding to the continuum, and the condition at r = r max yields no restriction at r max → ∞, whereas the condition at r = r 0 yields where K is a matrix (element of the Clifford algebra with generators α r , α ϕ , β) which obeys condition and without a loss of generality can be chosen to be Hermitian; in addition, it has to obey either condition [K, α r ] + = 0, or condition [K, α r ] − = 0.
One can simply go through four linearly independent elements of the Clifford algebra and find that two of them satisfy (18) and two other satisfy (19). However, if one chooses to satisfy (19), then (17) is violated. There remains the only possibility to choose with real coefficients obeying condition then both (17) and (18) are satisfied. Using obvious parametrization we finally obtain K = iβα r e −iθα r .
Thus, boundary condition (16) with K given by (23) is the most general boundary condition ensuring self-adjointness of the Hamiltonian operator on a surface with a deleted disc of radius r 0 , and parameter θ can be interpreted as the self-adjoint extension parameter. Value θ = 0 corresponds to the MIT bag boundary condition which was proposed as the condition ensuring the confinement of the matter field, that is, the absence of the matter flux across the boundary [16]. However, it should be comprehended that a condition with an arbitrary value of θ is motivated equally as well as that with θ = 0. Imposing the boundary condition (16) with matrix K (23) on the solution to the Dirac-Weyl equation, ψ E (x) (5), we obtain the condition for the modes: Let us compare this with the case of an infinitely thin (pointlike) disclination which was considered in detail in [11,12,13]. In the latter case several partial Hamiltonian operators are self-adjoint extended, and the deficiency index can be (0, 0) (no need for extension, all partial operators are essentially self-adjoint), (1, 1) (one partial operator is extended with one parameter), (2, 2) (two partial operators are extended with four parameters), etc. In particular, in the case of carbon nanocones, there is no need for self-adjoint extension for N d = 3, 4, 5, there is one self-adjoint extension parameter for N d = 2, 1, −1, −2, −3, −6, there are four and more self-adjoint extension parameters for N d = −4, −5 and N d ≤ −7. For the deficiency index equal to (1, 1), the boundary condition at the location of a pointlike disclination (r = 0) takes form where Θ is the self-adjoint extension parameter, F is given by (7) for N d = 2, 1, −1, −2, −3 and F = 1/2 for N d = −6, while n c = ± 1 2 [sgn(ν − 1) − 1] for N d = 2, 1, −1, −2, −3 and n c = ∓2 for N d = −6. As follows from the present section, in the case of a disclination of nonzero size, when the boundary condition is imposed at its edge, the total Hamiltonian operator is self-adjoint extended with the use of one parameter, see (24).

Quantum effects in the ground state of electronic excitations in nanocones
Using the explicit form of modes f ± n and g ± n , satisfying (6) and (24), we calculate current (10) and field strength (11). In the case of 3 5 < ν < 2 (0 < F < 1) we obtain and where ; (36) I ρ (u) and K ρ (u) are the modified Bessel functions with the exponential increase and decrease, respectively, at large real positive values of their argument.
The latter circumstance has far-reaching consequences, when we turn to the total flux of the induced ground state field strength, see (12). Namely, the contribution of the q-integral to Φ I is damped and the field strength is proportional to the current in the physically sensible case, i.e. at r max ≫ r 0 : where and

Conclusions
Quantum vacuum effects which are due to non-Euclidean geometry of nanocones are studied in the present paper. On the basis of the continuum model for longwavelength electronic excitations, originating in the tight-binding approximation for the nearest-neighbor interactions of the lattice atoms, we consider quantum groundstate effects in monolayers warped into nanocones by a disclination; the nonzero size of the disclination at the apex of a nanocone is taken into account. We show that the R-current circulating around the disclination is induced in the ground state, see (26) -(29) and (37). The pseudomagnetic field strength which is orthogonal to the nanocone surface is induced in the ground state as well, see (30) -(33) and (38). Both the current and the field strength are invariant under time reversal and consist of two parts: one is independent of the disclination size, r 0 , and another one depending on r 0 is damped at large distances from the disclination. In the physically sensible case, that is, at r max ≫ r 0 , the latter part is negligible, and we arrive at the conclusion that quantum ground-state effects are independent of r 0 . Moreover, in this case the field strength is proportional to the current, see (43), with Φ I being the total flux through a nanocone with radial size r max , see (44) -(47).
Our results are relevant for the two-dimensional Dirac materials of conical shape. In particular, for the case of the carbon monolayer (graphene) warped into a nanocone by a disclination, that is, a (6 − N d )-gonal (N d = 0) defect inserted in the otherwise perfect two-dimensional hexagonal lattice, the results can be summarized as follows. The dominating contribution to the induced ground-state flux of the pseudomagnetic field through carbon nanocones with N d = ±1, ±2, ±3, 4, 5, −6 is r max ∞ 0 du cosh 2 (u/2) sin 1 5 π cosh 9 10 u −sin 7 5 π cosh 3 10 u cosh 6 5 u − cos 6 5 π 5 7 π cosh 9 14 u +sin 1 7 π cosh 3 14 u cosh 6 7 u − cos 6 7 π , N d = −1, r max ∞ 0 du cosh 2 (u/2) sin 5 7 π cosh 15 14 u −sin 11 7 π cosh 3 14 u cosh 6 7 u − cos 6 7 π , N d = −1, Φ I = − e sin θ π r max , N d = ±2, −6, r max r max ∞ 0 du cosh 2 (u/2) cosh 5 6 u + cosh 1 6 u cosh 2 3 u − cos 2 3 π , N d = −3, We conclude that the quantum ground-state effects change drastically as N d changes. The effects are absent in the case of the four-heptagonal defect (N d = 4), whereas they appear of opposite signs as a heptagon is removed from (N d = 3) or added to (N d = 5) this defect, see (55) -(57). These cases are independent of the boundary parameter, θ; note that namely these cases correspond to that situation with the zero-size defect when there is no need for self-adjoint extension (the deficiency index is (0,0)). In all other cases the results depend on θ. The most distinct dependence is characteristic for the cases of two-pentagonal, two-and six-heptagonal defects, when the results coincide, see (52); note that the electric charge is not induced in these cases [11]. In the cases of one-pentagonal, one-and three-heptagonal defects, the results are almost independent of θ unless θ = − π 2 for N d = 1, −3 and θ = π 2 for N d = −1, see (48) -(51), (53) and (54). Freely suspended samples of crystalline monolayers are not exactly plane surfaces, but possesss ripples which produce pseudomagnetic fields causing strains and scattering of electronic excitations in a sample [17]. As follows from the present consideration, pseudomagnetic fields can be induced in the locally flat regions out of disclinations, and this may have observable consequences in experimental measurements, likely with the use of scanning tunnel and transmission electron microscopy.