Estructuración de un plan de negocios

The action in Eq. (4.1) has aU(1)symmetry which comes from the conservation of theJµ currents; this is boson charge conservation symmetry. It also has an SO(3) symmetry from the spins. Both of these symmetries participate in the protection of the topological phase, in the sense that if they are broken it is possible to continuously connect the topological phase to a trivial phase. In addition, the action has aZ2 symmetry, which is obtained by reflecting the~nspins in a plane in the spin space. To see how this affects the hedgehog current, we can examine Eq. (4.3), taking the reference vectorN~0to be in the plane of reflection. We see that reflecting the spins changes the sign of the imaginary part ofeiαµ_{, and therefore the hedgehog}

current changes sign under such a reflection. For our entire action to be invariant, we therefore need to combine such reflections with an operation which changes the sign of the boson currents. For concreteness

we will consider theZ₂symmetry corresponding to reflections of the~nvariables in theabplane of the spin space. In this case theZ₂symmetry can be summarized as:

na, nb → na, nb

nc → −nc

Qµ → −Qµ

Jµ → −Jµ

. (4.7)

Note that it is also possible to reflect the~nspins around a different plane, but this is not a distinct symmetry since it is simply the product of the aboveZ₂symmetry and an element ofSO(3).

By analogy with the electronic topological insulator, we would like theZ₂symmetry described above to be a “time-reversal” symmetry, i.e., it should be anti-unitary. The symmetry in Eq. (4.7) can be either a unitary or anti-unitary symmetry. Note that Eq. (4.1) is a real, (3+1)-dimensional action which is assumed to arise from the Trotter decomposition of the imaginary time propagator (i.e., Euclidean path integral) of a three- dimensional quantum Hamiltonian. The symmetry operations in Eq. (4.7) can be derived from the action of a symmetry operation on the quantum Hamiltonian. Therefore asking whether the symmetry in Eq. (4.7) is anti- unitary is the same as asking whether the symmetry of the quantum Hamiltonian which generates Eq. (4.7) is anti-unitary. This is a difficult question for us to answer as we do not know the quantum Hamiltonian which has Eq. (4.1) as its Euclidean path integral. Nevertheless, we take the perspective where we can check whether the original Hamiltonian has time reversal by complex-conjugating the action combined with the appropriate variable transformations, and in this way we can view the above symmetry of the action also as time reversal.

In the easy plane case, our system hasU(1)boson×U(1)spin in addition to this discrete symmetry. We can think of theU(1)spin as also coming from a boson, andna +inb gives the phase degree of freedom of this boson. We can imagine allowing tunnelling between the twoU(1)symmetries. If we do this, then the discrete symmetry should act the same way on the two species, and we can see that the above symmetry changes the number of the bosons but not their phase. In a system consisting of one species of bosons, the symmetry which acts in this way is anti-unitary. From now on the symmetry described above will also be treated as anti-unitary and denoted byZT₂.

The action in Eq. (4.1) is invariant under severalZ₂ symmetries, but for our purposes we will consider only theZT₂ symmetry described above as important, as it protects the topological behavior. To see this we can break this symmetry and argue that the topological phase is destroyed. We break theZT₂ symmetry by introducing a Zeeman field into our action:

SZeeman=−h

X

R

nc(R). (4.8)

breakU(1)spinorU(1)boson. In our picture of two species of vortices (Fig.4.2), the Zeeman field forbids one of the species. Since the vortex lines cannot change species, hedgehogs are forbidden and the binding phase is destroyed. This can be made more precise if we replace the binding term in Eq. (4.1) with the following term: λ 2 X r,µ [Jµ(r)−ηQµ(r)]2, (4.9)

whereηis a real number. If we choose the parametersβandλso that the system is initially in the binding phase, the introduction ofη allows us to tune the system between the binding phase (η = 1) and the trivial insulator (η = 0). Without a Zeeman field, the system undergoes a phase transition asηis changed between

1 and0. When a Zeeman field is applied, the hedgehogs are effectively forbidden, and so we can tune parametershandηwithout going through a phase transition. Indeed, whenη= 1, the change of variables in Eq. (4.6) leads to decoupledJ˜currents and spins, so there is no phase transition when makingharbitrarily large to align all spins. We can then tuneηto zero and finally reducehback to zero, all without undergoing a phase transition. Thus, in the presence of the Zeeman field, the phase withη = 1is not distinct from the trivial insulator.