7. TRABAJOS PREVIOS
10.5. Vigas prefabricadas de hormigón pretensado
By a similar argument the horizontal plane is not an obstruction to moving horizontalS1i-contours around and is in that sense trivial. We conclude that for the(ψ,φ,θ) = (π3, 0, 0)the horizontal winding numbers (m1, m2) are invariant
wherever they are defined.
Weyl lines in three dimensional Maxwellian lattices have already been de- scribed in the case of the generalized pyrochlore lattice [5], consisting of lines from the origin of the Brillouin zone to the origin of a Brillouin zone one recipro- cal lattice vector ahead. That is, these Weyl lines completely traverse a direction of the 3-torus. What is novel in the stacked kagome lattice, is the existence of closed loops that do not traverse the Brillouin zone, showing greater potential for shrinking them to the origin. We call these structures Weyl loops.
The most extensive Weyl loop found in our analysis is depicted in Fig.5.2, together with a visual proof of topological non-triviality. On the other end of the size spectrum we seem to find a series of parametrizations that produce arbitrarily small Weyl loops, as depicted in Fig.5.3. In the next chapter we will find that this shrinking can completely eliminate the Weyl loop.
ϕ=0.3
ϕ=0.5
ϕ=0.7
ϕ=0.8
ϕ=0.9
Figure 5.3: Sampling of the Weyl structures atψ= π/3andθ = 0for different values
ofφ. Asφgrows towards1≈π/3we see the Weyl loop shrink into the origin.
5.2
Edge signature
From Eq.(5.1) we read that in lattices with Weyl lines the winding numbers are no longer constant. In fact by consulting Fig.5.2b we see that by crossing the Weyl line the winding numbers will change by unity. As we know that the
36 Weyl lines
winding numbers encode the number of zero-modes on the right edge, we ex- pect a zero mode to flip from one to the other edge when crossing the Weyl line.
Figure 5.4: A zero mode changing edge. The Bril- louin zone is indicated by the cylindrical hull, with the axis and origin of the Brillouin zone indicated by the thin black lines.
This relation between the zero-energy edge modes and the zero-energy bulk modes is arguably best demonstrated by Fig.5.4, showing the complex plane from Fig.3.3 with an extra degree of freedom. At dif- ferent values of k1 we might find the corresponding
zero mode either at the top (|z3(k1)| < 1) or bottom
(|z3(k1)| > 1) edge, or in the bulk (|z3(k1)| = 1).
However, by continuity of these zero-energy struc- tures we know that if a mode changes side, it must pass through a non-trivial bulk zero.
Intuitively this can be understood by realizing that, as |z3| approaches 1 the penetration depth di-
verges, allowing it to hop between sides. A par- ticularly aesthetically pleasing example of this side switching is shown in Fig.5.5.
(a) (b)
Figure 5.5: Zero-energy edge modes bound to horizontal edges for the lattice corre- sponding to (ψ,φ,θ) = (π4,512π, 0). In a) all three bands are shown, while in b) the
middle band isolated with the Weyl loop drawn in green atλ = ∞. The vertical axis
indicates the inverse penetration depth1/λ = ln|zi|where negative (positive)λindi-
cates modes bound to the right (left) edge.
36
Chapter
6
Gapping the kagome lattice
6.1
Soft directions
One thing all the observed Weyl lines have in common is the fact that they form a connected path to the origin of the Brillouin zone. This suggests the following conjecture
Conjecture 1. All Weyl Lines are connected to the origin.
Figure 6.1: The linear ap- proximation (green) to the Weyl line sampling (blue) at (ψ,φ,θ) = (π3,π4, 0),
indicating the soft direc- tions. Only half of the Brillouin zone is shown.
This allows us to turn our attention to infinitesimal spheres around the origin. If our function is gapped there the conjecture guarantees that the function is gapped in the entire Brillouin zone. The advantage is that, instead of the polynomials of trigonometric functions involved in finding Weyl lines, the problem of finding these soft directions (see Fig.6.1) is a state- ment about shared real zeroes of low degree homo- geneous polynomials. The latter is computationally much more feasible, allowing us to sweep the com- plete phase space of triangle orientations, with suffi- cient resolution to distinguish the important features, overnight.
To investigate the neighbourhood of the origin, we
expand the determinant of the rigidity matrix aroundk =0. Recalling that the rigidity matrix depends on k only through the complex exponentials e±ιki we
can write down a general form for the expansion detQ(k) =
∑
n cne−ιn·k =∑
n∑
m (−ι)m m! cn(n·k) m =∑
m (−ι)m m! Pm(k), (6.1)38 Gapping the kagome lattice
wherePm(k) = cn(n·k)m are homogeneous polynomials of degreemwith real coefficients in three real variablesk1,k2andk3. The first three polynomials (the
constant, linear and quadratic terms) vanish identically. This is a reflection of the dimension of the kernel of Q(0) being at leastd = 3 (cf. Eq.(2.28)) and is proven in the appendix.
Since the polynomials are entirely real, we read from Eq.(6.1) that the real (complex) part of detQ(k) is given by summation over the even (odd) degree polynomials. If detQ(k) is to be zero somewhere on the infinitesimally small sphere around the origin, then both the lowest order real and the lowest order complex polynomial have to vanish there. That is, we must have a common real zero of the third and forth polynomial. This is made rigorous in the following lemma.
Lemma 1. If the rigidity matrix Q is zero along a path γthat connects to the origin,
then the corresponding third and fourth homogeneous polynomials satisfy
P3(kˆ) = P4(kˆ) = 0 where kˆ = dγ d||γ|| || γ||=0 .
Proof. As our path connects to the origin we can choose a parametrization such thatγ(0) = 0. Since this is a minimum of||γ||we can find a small interval[0,δ) on which||γ||is strictly increasing. On this interval we can re-parametrizeγby
its length:
γ: [0,||γ(δ)||)→R3: e 7→ eγˆe s.t. ||γˆe|| =1. (6.2)
Then we can use the homogeneity of the polynomials to find 0==detQ(γ(e)) =∑m≥1 (−1)m+1 (2m+1)!e 2m+1P 2m+1(γˆe) 0=<detQ(γ(e)) =∑m≥2 (−1)m (2m)!e2mP2m(γˆe) . (6.3)
We can find a very rough upper bound for the homogeneous polynomials over the unit sphere by
|Pm(γˆe)| =
∑
n cn(n·γˆe) m ≤∑
n |cn| |n·γˆe| m ≤ C(√ 3NB)m, (6.4) since|ni|is strongly bound by the number of rows ofQ, which in turn implies that the summation C = ∑n|cn| is finite. Rearranging Eq.(6.3) to express the lowest order homogeneous polynomial in the higher order polynomials we ob- tain the estimate|P3(γˆe)| ≤ 3! e3Cm
∑
≥2 e2m+1 (2m+1)!( √ 3NB)2m+1 |P4(γˆe)| ≤ 4! e4Cm∑
≥3 e2m (2m)! ( √ 3NB)2m , (6.5) 386.1 Soft directions 39
takinge =0 on both sides we obtain
P3(γˆ0) =0 and P4(γˆ0) = 0, (6.6) where ˆ γ0 =lim e→0 eγˆe e =lime→0 γ(e)−γ(0) e = dγ de e=0 . (6.7)
The logical inversion of this lemma, together with the conjecture, gives us the following theorem:
Theorem 2. If a system described by a rigidity matrix Q does not have any soft direc- tions, then the system cannot have Weyl structures.
Now what is left is a matter of letting the computer calculate the number of soft directions for different values of ψ, φand θ. We have plotted the result of such sweeps in figures Fig.6.2 and Fig.6.3. In the first figure we see that a region of phase space around ψ = φ = π/3 and θ = 0 do not have any soft directions. In the second figure we have plotted the number of soft directions corresponding to configurations withallthree angles non-zero. From this figure we observe that the zero-soft-direction region does not extend (deeply) into the θ 6= 0 regime. Furthermore, our investigation in the ψ,φ,θ 6= 0 cube indicates
0 π 12 π 6 π 4 π 3 5π 12 π 2 0 π 12 π 6 π 4 π 3 5π 12 π 2 ψ ϕ 0 π 12 π 6 π 4 π 3 5π 12 π 2 0 π 12 π 6 π 4 π 3 5π 12 π 2 ψ θ
Figure 6.2: Phase space plot showing the number of soft directions for different values ofψandφ(left) orψandθ(right), where the third angle is kept at zero. The colors rep-
resent the existence of 0 (blue), 2 (cyan), 4 (green), 6 (orange) or≥8 (red) soft directions at the origin of the Brillouin zone.
40 Gapping the kagome lattice
that soft-direction-less lattices are relatively rare, with our rough search return- ing the estimates for the soft-direction count distribution as expressed in Table 6.1.
This rareness makes it all the more exciting to find such a relatively large (though thin) region of parameter-space aroundψ,φ = π/3 andθ = 0, where the energy spectra of the corresponding lattices are completely devoid of any soft-directions at the origin, and thus devoid of Weyl structures.
Number of soft directions: 0 2 4 6 8 10 12
Percentage of phase space: <0.1 20.0 53.5 17.3 6.9 2.2 <0.1
Table 6.1: Estimate for the distribution of the number of soft directions over the cube where all three angle-parameters are strictly positive.
Figure 6.3:Phase space plot of configurations where all three angles are strictly positive. The seven images form a partition of the cube each showing those points with the same number of soft directions at the origin, with the number of soft directions included underneath.
0 2 4 6
8 10 12
40