However, if the path considered is cyclical, so thatP(0) =P(t), γn becomes gauge-invariant and, therefore, observable. Note that, in this case, the time-dependence is no longer essential.
It is the path traced out by the parameters, rather than their time dependence, that determines this quantity. Hence, it can be written
γn[Γ] =
˛
Γ
dPµAnµ = ˆ
Σ
dPµdPνΩnµν , (B.7)
where the second equality results from Stokes’ Theorem. This quantity is known as Berry (or geometric) phase. Here,Σis a surface in parameter-space such that Γ =∂ΣandΩnµν is the Berry curvature tensor, which is a gauge-invariant field or, put more succinctly, a gauge-field, reading
Ωnµν=∂µAnν −∂νAnµ . (B.8) This expression can be recast in a vector form, using a Levi-Civita symbol
Ωnµν =εµναΩnα⇔Ωnα= 1
3εµνα∂µAnν. (B.9)
The Berry phaseγn has the remarkable property that it may be proportional to an integer, i.e.
quantized. This happens when parameter-space is periodic in all coordinates and the surface Σis the whole parameter-space.
This result relies crucially on the fact that, although states along∂Σare identified, in virtue of periodicity, they can differ by a phase. Thus, there is a Berry phase accumulation when tracing a path over∂Σ, but this is not the phase of Eq. (B.5): this gauge dependent contribution vanishes, in virtue of periodicity. However, since the path is closed, the final state the path arrives at is the initial state. Since the phase must then be equal to unity, this imposes a condition on the Berry phase:
γn= 2πcn, cn ∈Z , (B.10)
wherecnis known as the first Chern number of then-th band. Note that we made no reference to the specificity of the physical system when arguing this result. Indeed, the only condition is the periodicity of the parameter space and, thus, of the Hamiltonian. This is, therefore, a characteristic of the topological structure of the mapping between the parameter-space spanned by{Pµ}and the states|n(P)i[111]. An intuitive but more detailed discussion of this result can be found in Ref. [111]. A more rigorous but accessible discussion can be found in Ref. [122].
Note that the Berry curvature, unlike the Berry phase, does not require the system to evolve along a closed loop in parameter-space in order to be a well-defined, observable quantity: it is an intrinsically local quantity, defined at each point in parameter-space. In this sense, the curvature is a more fundamental quantity than the phase.
Symmetries of the Berry curvature
Some important symmetries and conservation laws of the Berry curvature can be obtained directly from its definition. To see this, note that Eq. (B.8) can be rewritten as
−iΩnµν =∂µhn|∂νni −∂νhn|∂µni=h∂µn|∂νni − h∂νn|∂µni. (B.11) Taking the first equality, one can introduce a decomposition of identity such that
X
m6=n
|mihm|= ˆ1− |nihn|, (B.12)
and using the identity for the matrix element of the parameter-differentiated Hamiltonian hn|∂µHˆ |mi=δnm∂µεn+ (εn−εm)h∂µn|mi, (B.13) one obtains the following useful expression for the Berry curvature
Ωnµν =i X
m6=n
hn|∂µHˆ|mihm|∂νHˆ|ni − hn|∂νHˆ|mihm|∂µHˆ|ni
(εn−εm)2 . (B.14)
A conservation law becomes apparent from this expression. Rearranging terms and summing overnresults in an identically vanishing quantity:
X
n
Ωnµν ≡0. (B.15)
This is the local conservation law of the Berry curvature [111].
Consider now the spatial-inversion (parity) and time-reversal operators, denoted byPˆ and Tˆ, respectively, and a system in which the parametersPtransform, under bothPˆ andTˆ, as
OPˆ Oˆ†=−P, Oˆ= ˆP,Tˆ. (B.16) In order to determine the behaviour of the Berry curvature under these operations, we are going to use the elementary theorem that, if the Hamiltonian is invariant under a transformation, then the transformed eigenstates are also eigenstates of the Hamiltonian. As a corollary, if the energy spectrum is non-degenerate - as is the case we are considering - a symmetry operation maps an eigenstate into itself:
Oˆ|ni=|ni. (B.17)
There is a subtle point regarding the time-reversal operator that must be taken account of, which is that it is anti-unitary: for states|m0iand|n0iobtained through time-reversal of the states|mi and|ni, respectively, one has [109]
hm0|n0i=hm|Tˆ†T |niˆ =hm|ni∗=hn|mi. (B.18) We are now in a position to determine what we have set out to. We first consider a system with time-reversal symmetry. Thus, using the expression of Eq. (B.11), we find
−iΩnµν(P) = h∂µn|∂νni − h∂νn|∂µni
= (∂µ|ni)†(∂ν|ni)−(∂ν|ni)†(∂µ|ni)
(B.17)
=
∂µT |niˆ †
∂νT |niˆ
−
∂νT |niˆ †
∂µT |niˆ
(B.16)
= (−1)2
Tˆ∂µ|ni†
Tˆ∂ν|ni
−(−1)2
Tˆ∂ν|ni†
Tˆ∂µ|ni
= (∂µ|ni)†Tˆ†Tˆ(∂ν|ni)−(∂ν|ni)†Tˆ†Tˆ(∂µ|ni)
(B.18)
= h∂νn|∂µni − h∂µn|∂νni
= iΩnµν(−P)
⇔Ωnµν(P)=T −Ωnµν(−P).
For a system with spatial-inversion symmetry the proof follows similarly except for the sixth
equality, sincePˆ is simply a unitary operator [109]. Therefore, we have Ωnµν(P)= ΩP nµν(−P).
Far-reaching conclusions can be obtained from these results. First, a system with both spatial- inversion and time-reversal symmetry has an identically vanishing Berry curvature:
Ωnµν(P)=T −Ωnµν(−P)=P −Ωnµν(P)≡0. (B.19) Second, a system can only have a non-vanishing Chern number if the system breaks time- reversal symmetry without breaking spatial-inversion symmetry:
γn[∂Σ] = ˆ
Σ
dPµdPνΩnµν(P)
= (−1)2 ˆ
Σ
dPµdPνΩnµν(−P)
=T − ˆ
Σ
dPµdPνΩnµν(P)
=−γn[∂Σ]
≡0.
(B.20)
Moreover, although it may appear that the Berry curvature is merely an intermediate result in the computation of the Berry phase, that is only the case in a picture as the one used to derive Eq. (B.7), in which parameters evolve passively with time. Once the parameters are taken as dynamical variables, however, the Berry curvature reveals direct effects. In order to explore the deeper consequences of the Berry curvature in the dynamics of Bloch electrons, we must consider the perturbative effect of the remaining eigenstates on the evolution of the unperturbed eigenstaten. This amounts to taking the adiabatic approximation up to 1st order inP,tµ.
B.1.3 1st order adiabatic approximation
Recovering the adiabatic argument, we now expand the time-evolved quantum state|ψ(t)iin terms of the eigenstates|n(P(t))i, so that it can be written
|ψ(t)i=X
n
an(t)|n(P(t)), ti. (B.21) The equivalent to Eq. (B.2) under this gauge choice reads
∂t
eiγn(t)an(t)
=−X
m
eiγm(t)am(t)hn(t)|∂t|m(t)iexp
−i
~ ˆ t
0
dt0[εm(t0)−εn(t0)]
. (B.22)
Recovering the result of Eq. (B.3) to0-th order, we havea(0)n (t) = 1 anda(0)m(t) = 0, m 6= n, yielding
˙
a(1)n (t) =−iγ˙n(t)− hn(t)|∂t|n(t)i= 0⇒a(1)n (t) = 1,
∂t
eiγm(t)a(1)m (t)
=−eiγn(t)hm(t)|∂t|n(t)iexp
−i
~ ˆ t
0
dt0[εn(t0)−εm(t0)]
, m6=n , where, in the first line, Eq. (B.5) was used along with the relation∂t=∂tPµ∂µ. Integrating by parts the second line up to1-st order inQµ,t, we arrive at
a(1)m(t) =−i~hm(t)|∂t|n(t)i εn(t)−εm(t) expi
γn(t)−γm(t)−1
~ ˆ t
0
dt0[εn(t0)−εm(t0)]
+O(P,t2), (B.23) and inserting in the state of Eq. (B.21), we arrive at the time-evolved state of the quantum system up to1-st order it the adiabatic approximation:
|ψ1(t)in=X
j
a(1)j (t)|j(P(t)), ti+O(P,t2)
=eiγn(t)exp
−i
~ ˆ t
0
dt0εn(t0)
|n(t)i −i~ X
m6=n
hm(t)|∂t|n(t)i εn(t)−εm(t) |m(t)i
+O(P,t2). (B.24) This result is valid for any set of non-degenerate instantaneous eigenstates which depend on time through a set of parametersPµ. The indexnof the ket of the total state denotes that the initial state isnand, thus, its dominant component.
From now on, focus shall be given to a particular parameter-space: the Brillouin Zone.